Question

The snapshot graph at t=1 s and the history graph at x=1 cm of a wave are shown below (snapshot on left, history on right).

y(cn) ----1---------- ------ ---- 1 3 stn) -- -1 ----2-

1. What are the amplitude, frequency, and speed of the wave?

   Amplitude =    cm Frequency =    Hz    Speed = cm/s

2. What is the equation of this wave, expressed as y=f(x,y)? Express the coefficients of the x and t terms as decimals (i.e. don’t use π in your equation, use 3.14159...), and ignore units. Ignore the phase of the wave, and use sin, not cos. Also note that you are going to have to think pretty hard about which direction this wave is going.

Answer 1

(a) According to the graphs and their values indicated, the amplitude will be A = 2.5 cm

From y vs t graph the time period (T) is 2 s (time difference for one full sine wave). Therefore, the angular frequency, $\inline \omega$ = 2$\inline \pi$/T = 2$\inline \pi$/2 = $\inline \pi$ rad/s. The frequency, f = $\inline \omega$/2$\inline \pi$ = 0.5 Hz

From y vs x graph, the wavelength ($\inline \lambda$) is 2.5 cm (length of a full sine wave). Therefore, the speed of the wave will be v = f$\inline \lambda$ = 0.5 Hz x 2.5 cm = 1.25 cm/s

(b) The wave vector, k = 2$\inline \pi$/$\inline \lambda$ = 2$\inline \pi$/2.5 = 2.51327 cm^-1, and angular frequency, $\inline \omega$ = $\inline \pi$ rad/s = 3.14159 rad/s, The amplitude of the traveling wave, A = 2.5 cm. Therefore the equation for progressive wave is given by

$y\left ( x \right )=A\sin \left ( \omega t-kx \right )=2.5\sin \left ( 3.14159t-2.51327x \right )\ cm$