Do all standing waves have to have the same frequency and amplitude?
To get a pure standing wave, the two component travelling waves must have the same frequency and amplitude. You can experiment with different frequencies and amplitudes to see what you get.
Ooo hmm, interesting interesting. I am having trouble seeing what a non pure standing wave would look like. Ex. If one wave had a bigger amplitude, then when they construct it would have an amplitude inbetween, but when they destruct, then the amplitude might never go to zero at the anti node.
That's right - but I'll be a tad more careful: if one wave had a bigger amplitude than the other, their velocities were equal and opposite, and their wavelengths were the same, then the antinode does not go to zero - but to the difference between the two amplitudes. Will the two waves always interfere to produce fixed nodes though? $$y(x,t)= A\sin k(x-vt) + B \sin k(x+vt) = \left[A\sin k(x-vt) + A\sin k(x+vt)\right] + (B-A)\sin k(x+vt)$$... see what I did there? The part in square brackets has a solution you already know.
If the wavelengths are slightly different, but the amplitudes are the same, then you can get a complicated form of sloshing about like waves in a bathtub.
$$y(x,t)= A\sin k_1(x-vt) +A\sin k_2(x+vt)$$ ... you can see from the equation you can change a bit about. It's even possible for the wave-speeds to depend on the wavelength.
Note: ##k=2\pi /\lambda## so ##kv = 2\pi f = \omega##
Sounds like you are a visual thinker - so the algebraic approach tends not to work well for you: you need a mental picture?
You may have had an example of adding equal waves by hand, by taking snapshots at carefully chosen regular time intervals ... have a go doing that for the case that one wave has twice the amplitude of the other.
If you have access to some plotting software, you can use it to make an animation.
Someone has probably already done it ...
Thanks Simon Bridge! This was super helpful!
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