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<think>
Okay, let's see. The user wants an HTML file that animates the two-source interference pattern. Hmm, need to use a canvas. Alright, so first, I need to set up the canvas in the HTML. The main structure will be the HTML5 doctype, a body with a canvas element, and some JavaScript code.
The key parts are the animation loop with requestAnimationFrame. The two sources should be fixed but their distance varies over time. The waves expand from each source, and their amplitudes are summed at each point on the canvas. The color depends on the combined amplitude. Also, the distance between the sources must change slowly over time, and display that distance. Oh right, and restart with a new random instance when finished.
So, variables for the sources' positions. Maybe x1 and x2 for their x coordinates. The separation distance should be a variable that changes over time. Let's see, I need some parameters like wavelength, frequency, maybe the speed of the wave. Since both sources have the same frequency, their waves expand at the same speed.
The canvas needs to be drawn each frame. The waves can be modeled by tracking their radius over time. Since the animation is continuous, each frame increments the time. For each pixel (x, y) on the canvas, compute the distance to each source, then find the phase difference. The amplitude at that point would be the sum of the sine waves from both sources.
Wait, since the waves are expanding, the phase at a point (x,y) from source 1 would be (distance1 - speed*time)/(wavelength), then multiplied by 2π for the sine function. Similarly for source 2. The sum would be sin(...) + sin(...) which can be simplified using trigonometric identities, but maybe easier to compute directly in code.
The amplitude can be scaled to a color, perhaps using a grayscale where higher amplitude is brighter. The color can be set as a grayscale value based on the absolute value of the sum, to make sure it's between 0 and 1.
For the sources to move, the separation distance needs to be a function of time. Maybe start at a small separation and increase over time until some maximum, then reset? Or maybe oscillate between two values? Wait, the user said "slowly vary the distance over time so the fringe pattern visibly reshapes". So perhaps a linear increase over time until it reaches a maximum, then reset? Or a sinusoidal variation?
Alternatively, let the separation increase from zero to some value and then reset randomly. Hmm. The problem says "restart with a fresh random instance when finished". So maybe have the separation increase over a certain duration, then restart.
Wait, the animation is supposed to keep running, so perhaps it's a loop where after a certain period, it resets the sources to a new random position? Or maybe the separation varies in a cycle that restarts after a full cycle.
Alternatively, the separation could start at some value and slowly increase over time, and when it reaches a maximum, reset to a new starting point. But how to manage that in code?
Also, need to display the current separation distance. So a div element with an id that can be updated each frame.
Let me think of the variables:
- canvas width and height: maybe 600x400 for a decent size.
- sources: two points. Let's say their y-coordinate is in the middle of the canvas. So y = height / 2. The x positions can vary between some min and max.
- The separation distance is delta = x2 - x1. So each frame, we can adjust delta by adding a small delta increment. Wait, but how to make it vary smoothly over time until it reaches a point where we restart.
Hmm. Perhaps over a period of seconds, the separation increases, then when it reaches a certain point, reset to a new separation. Alternatively, have a period where the separation increases, then decreases, making a cycle.
Alternatively, just vary the separation between two values, so like from 100 to 300 pixels over 10 seconds, then go back to 100, etc. But the user said "slowly vary" so maybe a linear increase until it reaches a max, then reset. Let's see.
Alternatively, use a time variable that cycles, but that might complicate.
Wait, the problem says "Slowly vary the distance between the two sources over time...", so perhaps start at some initial separation, and each frame, add a small increment, but when it reaches a maximum, reset to a random value? Or a sinusoidal variation.
Maybe the separation is controlled by a time variable that is modulated. Let's have a variable that over time grows until it reaches a maximum, then resets. For example:
Let’s define a duration for the animation cycle. Let’s say 10 seconds. Over those 10 seconds, the separation increases from 50 to 250 pixels. Then after 10 seconds, it restarts. So each frame, we track the elapsed time, and compute the current separation based on that.
Alternatively, to make it continuous, maybe use a sinusoidal function where separation is something like 100 + 100 * sin(t / 5), so it oscillates between 0 and 200. Hmm, but that might be a way to vary it smoothly.
Alternatively, let’s just have the separation increase over time and when it reaches a certain value, reset to a new random start. Wait, but the user says "restart with a fresh random instance when finished". So maybe the animation runs for a certain period (like 10 seconds), then restarts with new parameters.
So here's a plan:
- The animation runs with a frame counter. Each frame, time increases. Once time reaches a maximum (like 10 seconds), we reset the sources' positions to new random locations, and reset the time.
Wait, but the problem states that the sources should be fixed for the duration of each "instance". So each instance runs until completion (until the separation reaches a point where it needs to reset), then a new instance starts with a new separation.
Hmm, perhaps the separation is set to start at a random value, then increases (or follows some path) over a certain duration before restarting.
Alternatively, each instance has the separation increasing from an initial value to a higher value over a period of, say, 10 seconds. After that, the instance restarts with a new initial separation and a new end separation, so each time it runs again, the path is different.
Alternatively, just have the separation vary in a way that over time it increases until a threshold, then resets. But how to code that.
Alternatively, for simplicity, have the separation be a function of time modulo a period. Let's pick a period T (like 20 seconds), and have the separation vary between some min and max over that period. For example, separation = minSep + (maxSep - minSep) * (sin((time / T)*2π) + 1)/2. Wait, that would make it oscillate between min and max.
Alternatively, let's set the separation as a linear function that increases over time until it reaches a maximum, then decreases, etc. But that might complicate.
Alternatively, just have the separation increase over 10 seconds, then reset. Let's say:
Initialize variables:
var startTime, separation = 50, targetSeparation = 200, duration = 10;
Then each frame, timeElapsed = (performance.now() - startTime)/1000;
separation = 50 + (200 -50)*(timeElapsed/duration); // linear increase from 50 to 200 over 10 seconds.
Once timeElapsed exceeds duration (10), we reset startTime to current time, and maybe change the target.
Wait, but to make it random each time:
Each time the animation restarts, choose new positions for the sources, so their separation is a new value. So when the time exceeds duration, reset the start time, pick new positions, etc.
Hmm, perhaps let's structure it as follows:
In the setup, we have variables that control the separation. When the animation starts, we set the two source positions to a random separation within a min and max range. Then, over a period (like 10 seconds), the separation smoothly increases to a larger random value. Wait, but the user said "slowly vary the distance between the two sources over time so the fringe pattern visibly reshapes".
Alternatively, the separation can be a sine wave function which creates a continuous variation. That might be simpler.
Let me think of code structure:
- The canvas is initialized with its context.
- The main animation loop uses requestAnimationFrame.
- The two sources are stored as objects with x and y positions. The y is fixed at canvas height / 2, so they're along the vertical center.
- The separation between them is controlled by a function of time.
- Each frame:
- Clear the canvas.
- Compute the current separation. Let's say the separation is a value that varies sinusoidally between 100 and 300 over, say, a 10-second period. So:
separation = 100 + 100 * (1 + Math.sin(time/5)); // time is in seconds, so over 10 seconds (since sin has 2π period), it would complete a full cycle every 10 seconds.
Wait, sin has a period of 2π, so if we have time/ (T/2π), where T is the period in seconds. For a 10-second period, the argument would be (time/(T/ (2π))) ?
Alternatively, let's set the separation as:
separation = 100 + 100 * Math.sin( (performance.now() - start)/ (1000 * 8) ) // 8 seconds period. Maybe that's okay.
But perhaps a better approach is to have the separation gradually increase over time, and when it reaches a maximum, reset to a random value again.
Alternatively, to make it cycle continuously, let's have the separation vary between two values in a smooth way.
Alternatively, let's just set it to vary sinusoidally between 50 and 250 over a period of 10 seconds. That would give a smooth variation and loop automatically.
So for the separation:
var t = (performance.now() - startTime) / 1000;
var separation = 150 + 50 * Math.sin(t * Math.PI * 2 / 10);
That way, every 10 seconds, the separation completes a full sine wave, varying between 100 and 200. So that's a smooth variation.
Alternatively, maybe start at some initial separation and slowly increase over time until it reaches a max, then reset. But to keep it simple, the sinusoidal approach might be easier.
Alright, moving on to the wave simulation.
The waves are circular, expanding from each source. For each frame, the time increases, so the radius of each wavefront is speed * time. But since both sources emit continuously, the waves are ongoing, not just a single pulse. So the phase at a point (x,y) due to source1 is (distance1 - speed*time)/lambda * (2π). But the wavelength and speed can be set such that the speed is wavelength * frequency. Since the frequency is same for both, the wavelength can be set arbitrarily.
Alternatively, since we need to compute the amplitude as the sum of the two waves, perhaps we can model the amplitude as:
amplitude1 = Math.sin(2 * Math.PI * (distance1 - speed * t)/lambda + phase1);
amplitude2 = Math.sin(2 * Math.PI * (distance2 - speed * t)/lambda + phase2);
Assuming both have same frequency (so same speed and wavelength), and same phase (since they start emitting at the same time?), then phase1 and phase2 can be zero. So the total amplitude is the sum of the two sines.
Alternatively, maybe the waves can be represented as:
The phase for each point from a source is (distance / wavelength) - (frequency * time). So the sine would be sin(2π*(distance/wavelength - frequency*time)).
But to prevent reflections, we don't need to worry about the grid edges.
The key is to compute for each pixel (x,y):
distance1 = Math.sqrt( (x - src1.x)^2 + (y - src1.y)^2 );
distance2 = similar for source2.
Then:
wave1 = Math.sin(2 * Math.PI * (distance1)/(wavelength) - (frequency * time) * 2 * Math.PI);
Wait, actually, the wave equation would be something like A * sin( (2π/λ)(r - vt) ) where v is the wave speed. Wait, r is the distance from the source, and vt is the speed multiplied by time. So the phase is proportional to (r - vt)/λ.
Alternatively, the general formula for a wave emitted from a point is of the form sin(k r - ω t), where k=2π/λ, ω=2π f.
So amplitude1 = sin(k * distance1 - ω * t). Similarly for amplitude2. Since both have the same frequency and wavelength, this should work.
Assuming that the speed v = ω/k = (2π f)/(2π/λ) )= f * λ. So speed is f * λ.
Therefore, the sum would be amplitude1 + amplitude2 = sin(k d1 - ω t) + sin(k d2 - ω t).
We can factor the expression as 2 sin( (k (d1+d2)/2 - ω t) ) cos( k (d1 - d2)/2 ), using the identity sin A + sin B = 2 sin( (A+B)/2 ) cos( (A-B)/2 )
Thus, the total amplitude simplifies to 2 sin( k (d1 + d2)/2 - ω t ) cos( k (d1 - d2)/2 )
The second term, cos( k (d1-d2)/2 ), depends on the path difference between the two sources to the point, so that's where the interference comes in.
The overall amplitude's maximum is 2 when the first sin term is 1, and the second term is 1, etc. So the absolute value of the sum can be up to 2. To map this to a grayscale, perhaps compute the absolute value and normalize to 0-255.
But in the code, perhaps it's better to compute the sum, take its absolute value, and then scale it for color.
Alternatively, just use the sum as the brightness, but since sine can be negative, we can take the absolute value.
Wait, the amplitude can be positive or negative, but the intensity is the square, but the problem says to color by combined amplitude, so maybe take absolute value.
Alternatively, use (amplitude1 + amplitude2) * 128 + 128, but clamp between 0-255. Or just take Math.abs(sum) and map it to a grayscale.
So for each pixel, after calculating the two sine terms, sum them, take absolute value, then scale it into 0-255 for grayscale.
Now, to implement this on a canvas:
We can loop through all pixels (x,y) on the canvas, compute distance to each source, then compute the sine terms summed, get the color.
But for a canvas of say 600x400, that's 240,000 pixels each frame. That might be too slow in JavaScript without optimizations.
Hmm, that's a problem. Because doing this in pure JS for 240k pixels each frame is going to be too slow and cause lag.
Wait, perhaps we can optimize by precomputing some variables or using typed arrays, but the user says no external libraries, so maybe just have to make do with what's possible.
Wait, perhaps using the 2D context's putImageData.
Alternatively, use the canvas's pixels as a buffer and compute each pixel's color by looping through all x and y.
But even so, for 600x400, that's 240,000 iterations per frame, which may be too slow unless optimized.
Hmm. Maybe reduce the resolution. Alternatively, make the grid not every pixel but every other pixel? Or use a smaller canvas. Let me think.
Alternatively, use a lower resolution, like 300x200, which reduces the number of pixels to 60,000. That might be manageable.
So let's set the canvas size to 600x400, but the drawing area is 600x400. Wait, but if we use the full size, each frame takes too long. Maybe using a smaller canvas like 300x200?
Alternatively, use a canvas with width and height set to 300 and 200, so the actual pixel count is lower.
Let me see: 300 * 200 = 60,000 pixels. Each pixel requires computing two distances and two sine terms. Maybe with some optimizations, that's manageable.
Alternatively, compute the two distances squared to avoid the square root? Not sure, but even sqrt is needed for the actual distance.
Hmm. Let me think of the code structure.
First, in the HTML:
<!DOCTYPE html>
<html>
<body>
<canvas id="c"></canvas>
<script>
// code here
</script>
</body>
</html>
We need to get the canvas context, set its width and height.
Then, variables for the sources:
var c = document.getElementById('c');
var ctx = c.getContext('2d');
var w = 600, h = 400;
c.width = w;
c.height = h;
var centerX = h / 2;
var src1X = 100, src2X = 200; // but these will be variable.
Wait, but the separation will be changing. Let's say the two sources are placed symmetrically around the center. So their positions can be:
var centerX = h / 2;
var separation = ...; // the current separation between them
var src1X = (w/2) - separation/2;
var src2X = (w/2) + separation/2;
Wait, but they're along the x-axis? Or along the y-axis? Wait, the problem says "circular" waves, so the sources are points in 2D space. Let's place them along the horizontal center line (y = h/2).
So their coordinates are:
src1 = {x: w/2 - separation/2, y: h/2}
src2 = {x: w/2 + separation/2, y: h/2}
So separation is the distance between the two sources.
Now, in the code, in each frame, we need to compute the current separation based on time.
But first, the variables:
var startTime = performance.now();
var wavelength = 20; // arbitrary value, affects how the waves look
var frequency = 1; // cycles per second
var speed = frequency * wavelength; // since v = f * λ
Wait, but using the parameters, the speed is f * λ. So with frequency 1, wavelength 20, speed is 20 units per second.
Alternatively, perhaps adjust these parameters for better visual effect.
Now, for each pixel (x, y):
distance1 = Math.sqrt( (x - src1.x)^2 + (y - src1.y)^2 );
distance2 = similar for source2.
Then the phases are:
phase1 = (distance1 / wavelength) - (frequency * time);
phase2 = (distance2 / wavelength) - (frequency * time);
amplitude1 = Math.sin( phase1 * 2 * Math.PI );
amplitude2 = Math.sin( phase2 * 2 * Math.PI );
sum = amplitude1 + amplitude2;
Then, the color is based on the absolute value of sum.
But to compute this for every pixel.
Alternatively, since the time is the same for all pixels, we can compute time once per frame.
time = (performance.now() - startTime) / 1000;
Then in the loops:
But even so, the loops are going to be slow.
Wait, perhaps we can vectorize the computations using canvas's methods, but I don't think that's possible without WebGL.
Alternatively, using a typed array for the image data and computing each pixel in a loop.
Here's an outline:
function draw() {
ctx.fillStyle = 'black';
ctx.fillRect(0,0,w,h);
var imageData = ctx.getImageData(0,0,w,h);
var data = imageData.data;
var t = (performance.now() - startTime)/1000;
// compute separation here, based on t and whatever function we choose
var separation = 150 + 50 * Math.sin(t * Math.PI * 2 / 10);
var src1 = {x: w/2 - separation/2, y: h/2};
var src2 = {x: w/2 + separation/2, y: h/2};
// loop over all pixels
for(var y=0; y<h; y++) {
for(var x=0; x<w; x++) {
// compute distances
var dx1 = x - src1.x;
var dy1 = y - src1.y;
var d1 = Math.sqrt(dx1*dx1 + dy1*dy1);
var dx2 = x - src2.x;
var dy2 = y - src2.y;
var d2 = Math.sqrt(dx2*dx2 + dy2*dy2);
// compute phases
var phase1 = (d1 / wavelength) - (frequency * t);
var phase1r = phase1 * 2 * Math.PI;
var a1 = Math.sin(phase1r);
var phase2 = (d2 / wavelength) - (frequency * t);
var phase2r = phase2 * 2 * Math.PI;
var a2 = Math.sin(phase2r);
var sum = a1 + a2;
var val = Math.abs(sum) * 128;
val = Math.min(255, Math.max(0, val));
// set the pixel to grayscale
var index = (y * w + x) *4;
data[index] = val;
data[index+1] = val;
data[index+2] = val;
data[index+3] = 255;
}
}
ctx.putImageData(imageData,0,0);
requestAnimationFrame(draw);
}
But this code is going to be very slow for a 600x400 canvas. For each frame, 240,000 pixels, each requiring two square roots and some sine calculations.
This might not run smoothly.
Possible optimizations:
1. Reduce the resolution by scaling down. For example, set the canvas width and height to 300x200, and then scale the canvas to double size, so the actual pixel computations are 300x200. This reduces the computation by a factor of 4.
2. Use typed arrays for the image data, but that's already happening.
3. Precompute some variables.
4. Approximate the distance calculation. For instance, skip the sqrt and use the squared distance, but that would change the phase.
Wait, but phase is proportional to distance divided by wavelength. So if we use the squared distance, that's not accurate. So that won't work.
Another idea: since the sources are moving, but in the current setup, they are moving sinusoidally between positions. Wait, but in the code above, the separation is changing, so the source positions are changing each frame.
Alternatively, maybe precompute separation every frame and keep the sources moving.
Hmm, perhaps this is too slow for a standard computer, but maybe with some optimizations.
Alternatively, we can skip every other pixel.
Alternatively, use a smaller canvas, say 300x200 or 200x150.
Let me adjust the canvas to 300x200:
c.width = 300;
c.height = 200;
Then the pixels would be 60,000, which might be manageable.
Another optimization: in the loops, instead of looping over x first, which might be slower, but not sure.
Also, precompute variables outside the loops where possible.
For example:
var frequency = 1;
var wavelength = 20;
var t = time;
var speed = frequency * wavelength;
But speed might not be used. Let me re-express the phases:
phase1 = (distance1 / wavelength) - (frequency * t);
But since wavelength = speed / frequency, so wavelength = (f * lambda) / f = lambda? Wait, perhaps this is getting confusing.
Alternatively, let's define parameters in terms of speed.
Let me see:
speed is the wave speed (units per second).
frequency is the number of cycles per second.
So wavelength = speed / frequency.
Thus, wavelength can be computed from speed and frequency.
But perhaps it's easier to just set speed = 20, frequency = 1, so wavelength is 20.
Alternatively, to make the animation reasonable, set the wavelength to something like 20 units, so that the waves are visible.
Another idea: precompute 1/wavelength, so that we don't have to divide in the inner loop.
var invWavelength = 1/wavelength;
Then phase1 = (d1 * invWavelength) - (frequency * t);
That's a minor optimization.
Also, precompute 2 * Math.PI once:
var tau = 2 * Math.PI;
Then phase1r = phase1 * tau;
Similarly for phase2.
Another optimization: precompute the sine terms using look-up tables, but that might be overkill.
Alternatively, compute the amplitude as Math.sin( (d1 * invWavelength - frequency * t) * tau ).
Perhaps combining terms:
var angle1 = (d1 / wavelength - frequency * t) * tau;
var a1 = Math.sin(angle1);
Same for a2.
Hmm.
Another optimization: in the distance computation, we can compute d1 squared and see if it's within some radius, but not sure.
Alternatively, precompute the center positions:
var centerX = c.width / 2; // 150 if width is 300
Wait, in code:
var w = 300, h = 200;
centerX = w/2 = 150
src1.x = 150 - separation/2;
src2.x = 150 + separation/2;
But separation is varying each frame.
Hmm.
Let me write sample code with these optimizations and a smaller canvas.
Testing the code:
Also, the separation function. Let's say separation varies between 50 and 150 over 10 seconds. So:
var separationBase = 100;
var separationRange = 50;
var separation = separationBase + separationRange * Math.sin( (performance.now() - startTime)*0.1 * Math.PI*2 / 10);
Wait, not sure.
Alternatively:
var sepTime = t; // t is seconds since start
var separation = 100 + 50 * Math.sin( sepTime * Math.PI * 2 / 10 );
This would make separation oscillate between 50 and 150 every 10 seconds.
Alternatively, start at 100 and vary between 50 and 150.
Thus, the separation is:
100 + 50 * Math.sin( (t) * (2*Math.PI)/10 )
Thus, every 10 seconds, it completes a sine wave.
This way, separation varies smoothly between 50 and 150 every 10 seconds.
Now, with all these parameters, the code might be manageable with a smaller canvas.
Testing in code:
Let me code this with a 300x200 canvas:
// Set up the canvas
var canvas = document.getElementById('myCanvas');
var ctx = canvas.getContext('2d');
canvas.width = 300;
canvas.height = 200;
var startTime = performance.now();
// Parameters
var wavelength = 20; // units per cycle
var frequency = 1; // cycles per second
var invWavelength = 1 / wavelength;
var tau = 2 * Math.PI;
var w = canvas.width;
var h = canvas.height;
var centerX = w / 2;
var centerY = h / 2;
function draw() {
var t = (performance.now() - startTime) / 1000;
// Compute separation
var sepTime = t;
var separation = 100 + 50 * Math.sin( sepTime * Math.PI * 2 / 10 );
var src1X = centerX - separation / 2;
var src2X = centerX + separation / 2;
// Get image data
ctx.fillStyle = 'black';
ctx.fillRect(0,0,w,h);
var imageData = ctx.getImageData(0,0,w,h);
var data = imageData.data;
// Precompute some variables
var freqT = frequency * t;
// Iterate over all pixels
for(var y=0; y<h; y++) {
for(var x=0; x<w; x++) {
// Compute distances to each source
var dx1 = x - src1X;
var dy1 = y - centerY;
var d1Squared = dx1*dx1 + dy1*dy1;
var d1 = Math.sqrt(d1Squared);
var dx2 = x - src2X;
var dy2 = y - centerY;
var d2Squared = dx2*dx2 + dy2*dy2;
var d2 = Math.sqrt(d2Squared);
// Compute phases
var phase1 = (d1 * invWavelength) - freqT;
var angle1 = phase1 * tau;
var a1 = Math.sin(angle1);
var phase2 = (d2 * invWavelength) - freqT;
var angle2 = phase2 * tau;
var a2 = Math.sin(angle2);
var sum = a1 + a2;
var val = Math.abs(sum) * 128; // scale to 0-255
val = val > 255 ? 255 : (val <0 ? 0 : val);
// set pixel color
var index = (y * w + x)*4;
data[index] = val;
data[index+1] = val;
data[index+2] = val;
data[index+3] = 255;
}
}
ctx.putImageData(imageData,0,0);
requestAnimationFrame(draw);
}
draw();
But this is still going to be slow because of the nested loops. To make this manageable, perhaps use a web worker or use typed array optimizations.
Alternatively, precompute the distance fields for each source and then combine them.
Alternatively, use WebGL.
But for the purpose of creating a simple HTML canvas animation, perhaps this can be optimized further.
Another idea: vectorize calculations using array methods or webgl.
Alternatively, using a larger wavelength to reduce the number of cycles, making the math simpler.
Alternatively, precompute the separation and source positions before entering the loops.
Wait, in the inner loops, the separation and source positions are fixed per frame, so precomputing them outside the loops is necessary.
But even so, the nested loops are going to be slow.
Possible optimization: compute the distance squared and avoid sqrt, but that changes the result.
Wait, phase is proportional to distance divided by wavelength. If we use distance squared divided by wavelength, that would be different.
Alternatively, approximate the distance as the max(dx, dy). But not sure.
Alternatively, use integer calculations where possible.
Alternatively, use typed arrays for the data buffer.
But in the code above, we already are using typed arrays via getImageData.
Hmm.
Alternatively, use a single loop:
for(var i=0; i<data.length; i+=4) {
But then, we'd have to compute x and y from the index, which might be slower.
Alternatively, compute x and y in a way that's efficient.
Wait, in the current code, the inner loop is over x, which is the fastest changing index. So in the data array, consecutive pixels are stored in order, so the inner loop over x is the best way.
Hmm.
Alternatively, precompute the coordinates in a separate step.
Wait, perhaps try to run this code and see if it is too slow. If it is, maybe reduce the canvas size further.
Alternatively, use a 200x150 canvas.
Alternatively, let me try with 150x100.
canvas.width = 150;
canvas.height = 100;
Then the total pixels are 15,000, which is manageable.
Alternatively, see if there's a better way.
Alternatively, compute the amplitude for all pixels using a mathematical formula that can be vectorized.
Wait, the problem is that each pixel requires two distance calculations to two moving points, then the sine functions.
Alternatively, precompute for each pixel the position once, then just compute the distance each frame.
But that's not feasible because the sources are moving each frame.
Hmm.
Another Idea: since the sources are moving sinusoidally, their positions can be represented as functions of time. Let me see:
The separation between the two sources is:
separation(t) = 100 + 50 * sin( (t / 10) * 2π )
Thus, the position of each source is:
x1(t) = centerX - separation(t)/2
x2(t) = centerX + separation(t)/2
Thus, the distance from a pixel (x,y) to each source is:
d1 = sqrt( (x - x1)^2 + (y - centerY)^2 )
Similarly for d2.
Perhaps precompute the centerY as h/2 = 100 (if h=200).
Wait, in the code above with h=200, centerY is 100.
Wait, in the code I wrote earlier, centerY was set to h/2 but in the code:
var centerY = h / 2;
Wait yes.
Thus, for each pixel (x,y), the y-coordinate is fixed as it's along the horizontal center line for the sources.
Hmm, but the y-coordinate of the sources is centerY, so the distance includes the vertical component.
Wait, in this setup, the sources are along the horizontal center line (y = centerY), so the vertical distance from any pixel to the sources is (y - centerY). Thus, the distance squared is (dx)^2 + (dy)^2, where dy = y - centerY.
Thus, perhaps precompute the dy term once per row.
For each row y, we can precompute dy = y - centerY.
Then, for each x in that row, compute dx1 = x - x1, dx2 = x - x2.
Thus, in the loops:
for each y in 0..h-1:
dy = y - centerY;
dySquared = dy*dy;
for each x in 0..w-1:
dx1 = x - x1;
d1Squared = dx1*dx1 + dySquared;
d1 = sqrt(d1Squared);
similarly for dx2, d2...
This way, we avoid recomputing dy squared for every pixel, which saves some computation.
But still, the sqrt is expensive.
Alternatively, precompute the 1/wavelength and other terms outside.
Hmm.
Alternatively, use an approximation for sqrt.
But that might introduce inaccuracies.
Alternatively, precompute all the necessary distances in a helper array, but that might not help.
Alternatively, use look-up tables for sine functions, but again, not sure.
Alternatively, note that Math.sin is a built-in function and optimized, so maybe not too bad.
Hmm.
Perhaps the best way is to proceed with the code written above, using a smaller canvas (e.g., 200x150) and see if it runs smoothly.
Alternatively, let me write the code with these optimizations and parameters:
Here's the revised code with a 200x150 canvas:
```javascript
var canvas = document.getElementById('myCanvas');
var ctx = canvas.getContext('2d');
canvas.width = 200; // smaller dimensions
canvas.height = 150;
var startTime = performance.now();
// Parameters
var wavelength = 20; // units per cycle
var frequency = 1; // cycles per second
var invWavelength = 1 / wavelength;
var tau = Math.PI * 2; // 2 * Math.PI
var w = canvas.width;
var h = canvas.height;
var centerX = w / 2;
var centerY = h / 2;
function draw() {
var t = (performance.now() - startTime) / 1000;
// Compute separation
var sepTime = t;
var separation = 100 + 50 * Math.sin( sepTime * tau / 10 );
var src1X = centerX - separation / 2;
var src2X = centerX + separation / 2;
// Clear the canvas
ctx.fillStyle = 'black';
ctx.fillRect(0, 0, w, h);
// Get image data
var imageData = ctx.getImageData(0, 0, w, h);
var data = imageData.data;
// Precompute some variables
var freqT = frequency * t;
// Iterate over all pixels
for(var y=0; y<h; y++) {
var dy = y - centerY;
var dySquared = dy * dy;
for(var x=0; x<w; x++) {
// Compute dx to each source
var dx1 = x - src1X;
var d1Squared = dx1 * dx1 + dySquared;
var d1 = Math.sqrt(d1Squared);
var dx2 = x - src2X;
var d2Squared = dx2 * dx2 + dySquared;
var d2 = Math.sqrt(d2Squared);
// Compute phases
var phase1 = (d1 * invWavelength) - freqT;
var angle1 = phase1 * tau;
var a1 = Math.sin(angle1);
var phase2 = (d2 * invWavelength) - freqT;
var angle2 = phase2 * tau;
var a2 = Math.sin(angle2);
var sum = a1 + a2;
var val = Math.abs(sum) * 128; // Scale to 0-255
// Clamp the value between 0 and 255
val = val > 255 ? 255 : (val < 0 ? 0 : val);
// Set pixel color
var index = (y * w + x) * 4;
data[index] = val;
data[index + 1] = val;
data[index + 2] = val;
data[index + 3] = 255;
}
}
ctx.putImageData(imageData, 0, 0);
requestAnimationFrame(draw);
}
draw();
```
This should be more optimized, but still, the sqrt operations and nested loops might be slow.
Alternatively, using WebGL would be better for performance, but since the user asked for a canvas creation, we proceed with this.
Another optimization: precompute the separation and source positions outside the loops.
Wait, in the above code, that's already done.
Perhaps using typed arrays and precomputing some constants can help.
Alternatively, precompute the values for each row y once, then compute for each x.
Alternatively, note that the equation is symmetric vertically, but since the sources move horizontally, symmetry might not help.
Alternatively, use a pixel buffer and manipulate it as an array.
Hmm.
Perhaps with the reduced canvas size (200x150 = 30,000 pixels), this might work, albeit not very fast.
Alternatively, the user might need to implement this in a more efficient way, but given the constraints, this is a possible answer.
Final answer:
Here's the JavaScript code that creates the animation described, using a canvas with a smaller size for better performance: