00:01
This problem is a wave equation.
00:03
And by the wave equation, i don't mean, you know, y of x, t is equal to a cosine kx minus omega t.
00:13
No, that is the equation that describes a wave, but the wave equation i'm talking about right now is this equation here, which is a partial differential equation, which all waves must satisfy.
00:25
So it says if you have some function y of x comma t, then it is a wave if it satisfies this partial differential equation.
00:33
So we want to know if some of these functions that the textbook gives you are waves or not.
00:40
So first up is y of x comma t is equal to a cosine kx plus omega t.
00:48
Now we already know this is a wave because we've worked with this and other problems, but let's go ahead and verify it.
00:54
So all we have to do to verify this is take d squared y by dx squared and d squared y by dt squared and see if they satisfy this relationship.
01:05
So d squared y by dx squared, hopefully you're not too rusty with your derivatives, is getting me equal to minus a k squared times cosine of kx plus omega t.
01:24
And d squared y by d t squared is going to be equal to minus a omega squared cosine of kx plus omega t and so here's the thing as long as v is equal to omega k then it is satisfied and i hope you see now why we have the relationship v is equal to omega over k, or equivalently v is equal to lambda f.
02:13
And that's because when you write the equation in this form and you plug them into the wave equation, it gives you an equation for what v is.
02:22
So i just think it's interesting to see how the math works out with that.
02:25
And so you can understand through using these derivatives, why v is equal to omega over k.
02:30
And so next up is y of x and t.
02:34
Is a sine x, sine kx plus omega t...