The wave function y(x,t) for a one-dimensional transverse wave: equation for a moving wave.
00:00 We develop the mathematical description of a wave: for a one-dimensional transverse wave, the wave function y(x,t) is a function of two variables because the height of the string depends on both position and time. We begin with an animation of a one dimensional wave moving to the right, then we highlight a spot on the string and establish that as the wave moves to the right, a spot on the string moves perpendicular to the direction of wave velocity. This is why it's called a transverse wave! Finally we review some basic wave vocabulary by labeling a diagram of the wave with amplitude, wavelength and wave speed.
00:45 Dependence on x: we freeze the animation and describe the static wave as y(x)=Acos(kx) where k is the wave number which determines the wavelength. We quickly establish the connection to wavelength: lambda=2pi/k, that's the change in x required to complete one cycle of the wave with respect to x.
02:18 Dependence on t: we use a function translation, replacing x with x-d to shift the wave by a distance of d to the right. Since the wave is moving at constant velocity, we can say d=vt, and we have the time dependence of our wave function, the equation for a moving wave y(x,t)=Acos(k(x-vt)). Distributing k and replacing kv with omega (the angular velocity), we obtain the equation of the wave function y(x,t)=Acos(kx-omega*t), the standard equation for a wave moving to the right. We get the simple formula for wave velocity omega/k, the angular frequency divided by the wave number.
03:53 For a fixed value of x, we observe a spot on the string moving in simple harmonic motion. The period of this motion is given by 2pi/omega, so the frequency is omega/2pi and we can also say omega=2pi*f. Now we can relate the formula for wave speed = omega/k to the formula we derived long ago: v=f*lambda, and show they are the same!
05:30 Leftward moving waves: finally, we show an animation of a leftward moving wave and indicate how the derivation would change: the translation would simply be leftward instead of rightward, using a plus sign instead of a minus. Thus, the wave equation for a wave moving to the left is y(x,t)=Acos(kx+omega*t).
