Wave function for a standing wave, boundary conditions on a string of length L, normal modes, etc.
00:00 When two waves of the same frequency run into each other on a string, the interference of the rightward moving wave and leftward moving wave produces a standing wave. The standing wave appears to stand still, and there are some points that are fixed in place at the equilibrium position. These are called the nodes of the standing wave. In between the nodes, we find points of maximum amplitude on the standing wave, and these are called the antinodes. Our first goal in this video is to find the explicit wave function for a standing wave, then we're going to find out which wavelengths are allowed for standing waves on a string clamped at both ends, where the waves are interfering with their own reflections (and we'll visualize those standing wave animations). Finally we'll write down the allowed frequencies corresponding to those wavelengths, in other words the resonant frequencies or harmonics for the string.
00:36 Combining the leftward and rightward moving waves: We begin by writing down the wave equation for a wave moving to the right: y(x,t)=Acos(kx-wt) and a wave moving to the left: y(x,t)=Acos(kx+wt). Since our leftward moving waves are reflections of the rightward moving waves, we need to adjust them with a minus sign, so that's y(x,t)=-Acos(kx-wt). Now we add the waves and apply the trig identities for the cosine of a sum and difference to obtain the wave function for a standing wave: 2Asin(kx)sin(wt). Note that the amplitude of the standing wave is twice the amplitude of the original waves that we combined to get the standing wave!
03:21 Standing wave boundary conditions on a string of length L, normal modes: For a string fixed at both ends, we require that y(0,t)=0 and y(L,t)=0, in other words the vertical displacment is always zero at the ends of the string. Our wave function for the standing wave already meets the first boundary condition since the sine of zero is zero. The second boundary condition requires that kL=n*pi, so we obtain formulas for the nth allowed wave number and thus the nth allowed wavelength. The waves corresponding to each n are called normal modes. We note that every choice of n determines a resonant wavelength on the string, but it also changes the resonant frequency: as n gets larger, we have smaller wavelengths and larger frequencies, such that the wave velocity remains constant for all the normal modes.
05:30 Animations of the standing waves: we plug in n=1,2,3,4 and find the corresponding wavelengths and wave functions for the first four standing waves on a string of length L. For each one, we note that the frequencies of oscillation are growing faster!
06:56 Frequencies of harmonics on the string: finally, we investigate the resonant frequencies on the string. We use the wave speed formula to write down the nth harmonic frequency in terms of wavelength, then we apply the wavelength=2L/n formula to write the nth frequency as nv/2L. Noting that f_1 =v/2L, we substitute back into the previous formula to obtain f_n=nf_1, in other words every resonant frequency is a multiple of the fundamental frequency or first harmonic. This is one of the most powerful formulas for problem solving with standing waves!
