A question about stationary waves.
I've learnt that a stationary wave is formed after two waves of equal frequency (but not necessarily equal amplitude) travelling in opposite directions superpose.
Now, the textbook says that because they have the same frequency, at certain points there are antinodes and nodes (points of maximum amplitude and zero amplitude respectively). It says that these points are formed due the waves having the same frequency and hence they are in antiphase and in phase at these points.
But I thought what if the frequencies of the waves were equal, but the wavelengths weren't?
Would there still be antinodes and nodes?
Now, the textbook says that because they have the same frequency, at certain points there are antinodes and nodes (points of maximum amplitude and zero amplitude respectively). It says that these points are formed due the waves having the same frequency and hence they are in antiphase and in phase at these points.
But I thought what if the frequencies of the waves were equal, but the wavelengths weren't?
Would there still be antinodes and nodes?
I don't believe you'd still have nodes and anti-nodes.
Roughly speaking, by assuming an equal amplitude, it seems easy to show that assuming an equal frequency there will be nodes/antinodes.
If the amplitudes aren't equal, but the frequencies are, I'm unsure about whether it'd be the case, I can't prove that mathematically.
The only condition I can actually show that for is if the amplitudes are the same and they have the same angular frequency.
Consider:
1) sin(k1*x - omega*t)
2) sin(k2*x + omega*t)
Two waves, opposite directions (omega)
Superposition with equal amplitude
sin(0.5*sum)cos(0.5*difference) ->>>
sin((0.5)(k1 + k2)x)cos(etc)
There exist values of (x) where the superposition of the waves equals zero.
Now if the amplitudes aren't equal, that's all fudged, and frankly I don't think I can sum up like that. There're bound to be... HOLY THAT CAT IS SO CUTE, anyway, there are times at which there might be a zero, but there (I can't see it mentally) won't be nodes and antinodes.
The only way this works appears to be if the amplitudes are equal, otherwise there won't exist any nodes or antinodes.
Correct me if I'm wrong someone, this is some pretty dirty math and the addition formula might be fudged.
Roughly speaking, by assuming an equal amplitude, it seems easy to show that assuming an equal frequency there will be nodes/antinodes.
If the amplitudes aren't equal, but the frequencies are, I'm unsure about whether it'd be the case, I can't prove that mathematically.
The only condition I can actually show that for is if the amplitudes are the same and they have the same angular frequency.
Consider:
1) sin(k1*x - omega*t)
2) sin(k2*x + omega*t)
Two waves, opposite directions (omega)
Superposition with equal amplitude
sin(0.5*sum)cos(0.5*difference) ->>>
sin((0.5)(k1 + k2)x)cos(etc)
There exist values of (x) where the superposition of the waves equals zero.
Now if the amplitudes aren't equal, that's all fudged, and frankly I don't think I can sum up like that. There're bound to be... HOLY THAT CAT IS SO CUTE, anyway, there are times at which there might be a zero, but there (I can't see it mentally) won't be nodes and antinodes.
The only way this works appears to be if the amplitudes are equal, otherwise there won't exist any nodes or antinodes.
Correct me if I'm wrong someone, this is some pretty dirty math and the addition formula might be fudged.
Original post by Callicious
I don't believe you'd still have nodes and anti-nodes.
Roughly speaking, by assuming an equal amplitude, it seems easy to show that assuming an equal frequency there will be nodes/antinodes.
If the amplitudes aren't equal, but the frequencies are, I'm unsure about whether it'd be the case, I can't prove that mathematically.
The only condition I can actually show that for is if the amplitudes are the same and they have the same angular frequency.
Consider:
1) sin(k1*x - omega*t)
2) sin(k2*x + omega*t)
Two waves, opposite directions (omega)
Superposition with equal amplitude
sin(0.5*sum)cos(0.5*difference) ->>>
sin((0.5)(k1 + k2)x)cos(etc)
There exist values of (x) where the superposition of the waves equals zero.
Now if the amplitudes aren't equal, that's all fudged, and frankly I don't think I can sum up like that. There're bound to be... HOLY THAT CAT IS SO CUTE, anyway, there are times at which there might be a zero, but there (I can't see it mentally) won't be nodes and antinodes.
The only way this works appears to be if the amplitudes are equal, otherwise there won't exist any nodes or antinodes.
Correct me if I'm wrong someone, this is some pretty dirty math and the addition formula might be fudged.
Roughly speaking, by assuming an equal amplitude, it seems easy to show that assuming an equal frequency there will be nodes/antinodes.
If the amplitudes aren't equal, but the frequencies are, I'm unsure about whether it'd be the case, I can't prove that mathematically.
The only condition I can actually show that for is if the amplitudes are the same and they have the same angular frequency.
Consider:
1) sin(k1*x - omega*t)
2) sin(k2*x + omega*t)
Two waves, opposite directions (omega)
Superposition with equal amplitude
sin(0.5*sum)cos(0.5*difference) ->>>
sin((0.5)(k1 + k2)x)cos(etc)
There exist values of (x) where the superposition of the waves equals zero.
Now if the amplitudes aren't equal, that's all fudged, and frankly I don't think I can sum up like that. There're bound to be... HOLY THAT CAT IS SO CUTE, anyway, there are times at which there might be a zero, but there (I can't see it mentally) won't be nodes and antinodes.
The only way this works appears to be if the amplitudes are equal, otherwise there won't exist any nodes or antinodes.
Correct me if I'm wrong someone, this is some pretty dirty math and the addition formula might be fudged.
What about the wavelengths?
Original post by optalk
What about the wavelengths?
The wavenumber/vector/etc represents them. When you represent a wave, it's usually of the sinusoid form (for this case) of
Asin(kx - wt)
Where w = 2*pi*F and k = 2*pi*1/(lambda)
w/ lambda = wavelength.
When amplitude is equal, it works out fine, as I've shown. If not, then I don't think it's possible to get a standing wave, even with the same frequency.
It might be possible, but I can't see how myself. Someone else will have to take up the torch for that.
There would be antinodes and nodes but it wouldn’t be a stationary wave. If the frequency is the same but wavelength isn’t then the velocities would be different. This would result in antinode and node positions shifting.
Edit: make sure this is checked.
Edit: make sure this is checked.
(edited 5 years ago)
Dude u to make me want to go revise physics I’ve been procrastinating
Original post by Vikingninja
There would be antinodes and nodes but it wouldn’t be a stationary wave. If the frequency is the same but wavelength isn’t then the velocities would be different. This would result in antinode and node positions shifting.
This is what I mean, does the markscheme expect us to assume that the waves are identical? In the textbook, it doesn't state whether or not the wavelengths can be different. It just says the frequencies have to be the same.
Original post by someusernamet
Dude u to make me want to go revise physics I’ve been procrastinating
Physics is fun bro. Some concepts are just difficult to grasp.
Yh ik the difficult concepts are fun too tho
Original post by optalk
This is what I mean, does the markscheme expect us to assume that the waves are identical? In the textbook, it doesn't state whether or not the wavelengths can be different. It just says the frequencies have to be the same.
If you can would check this with your teacher. Been a good few years since I did this topic.
Original post by Vikingninja
If you can would check this with your teacher. Been a good few years since I did this topic.
I can't, it's easter holidays and I don't want to wait.
Original post by Callicious
Not going to lie, this is way ahead of me and I 110% don't envy you for having to learn all of that math.
Original post by optalk
Not going to lie, this is way ahead of me and I 110% don't envy you for having to learn all of that math.
They teach that for AS/Math/Physics,
It's just applying the sine addition formulae and the sinusoid wavefunction that you're given in A-Level;
Anyhow,
It's just what I said earlier in picture form. Seems like it'd be useful ;-;
Original post by Callicious
They teach that for AS/Math/Physics,
It's just applying the sine addition formulae and the sinusoid wavefunction that you're given in A-Level;
Anyhow,
It's just what I said earlier in picture form. Seems like it'd be useful ;-;
It's just applying the sine addition formulae and the sinusoid wavefunction that you're given in A-Level;
Anyhow,
It's just what I said earlier in picture form. Seems like it'd be useful ;-;
It's not in the A1 textbook for physics and it's way ahead in what we're learning in pure maths right now. Don't want to befuddle my brain.
I know it had been a month. Hope the following writing can provide some insight to the questions posed by OP.
The textbook that mentioned the above, is it endorsed by the exam board? If not, I would advise you to change your textbook.
I have attached a graph that stimulates the superposition of 2 progressive waves (dotted lines) travelling in opposite directions with unequal amplitudes below. The resultant waves at various times are solid lines.
Do the resultant waves have the features of a standing wave? I don’t think.
No.
Original post by optalk
I've learnt that a stationary wave is formed after two waves of equal frequency (but not necessarily equal amplitude) travelling in opposite directions superpose. …
The textbook that mentioned the above, is it endorsed by the exam board? If not, I would advise you to change your textbook.
I have attached a graph that stimulates the superposition of 2 progressive waves (dotted lines) travelling in opposite directions with unequal amplitudes below. The resultant waves at various times are solid lines.
Do the resultant waves have the features of a standing wave? I don’t think.
Original post by optalk
… But I thought what if the frequencies of the waves were equal, but the wavelengths weren't?
Would there still be antinodes and nodes?
Would there still be antinodes and nodes?
No.
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