Question

(1) Using an appropriate trig identity or two (sin (A + B), sin (A + π), sin A+ sin B, eg), write an expression for a superposition of the following waves. (a) Two waves with amplitude 2 meters overlap. They both have zero phase shift, and they are traveling in the same direction. One of them has a wavelength of 3 meters and freqIKxey 5 11%. The oi,her has a wavelculnih of 5 1ncion; and 3 Ik. (b) Two waves with amplitude 5 meters overlap. One has a phase shift of π relative to the other one, and they are traveling in the same direction. They both have a wavelength of 3 meters and frequency 5 Hz.

Answer 1

The general equation of a travelling wave is given by -

y (x, t) = A sin (k x - $\omega$ t + _$\phi$)

For both the waves, we have

_$\phi$ = phase shift = 0

We know that, k = wavenumber = 2$\pi$ / $\lambda$      and $\omega$ = angular frequency = 2$\pi$f

For a first wave, we have

y₁ (x, t) = A sin (k₁ x - $\omega$₁ t)

y₁ (x, t) = (2 m) sin { [(6.28 rad) / (3 m)] x - [(6.28 rad) (5 Hz)] t }

y₁ (x, t) = (2 m) sin [(2.093 m^-1) x - (31.4 rad/s) t]

For a second wave, we have

y₂ (x, t) = A sin (k₁ x - $\omega$₁ t)

y₂ (x, t)= (2 m) sin { [(6.28 rad) / (5 m)] x - [(6.28 rad) (3 Hz)] t }

y₂ (x, t) = (2 m) sin [(1.256 m^-1) x - (18.84 rad/s) t]

An expression for a superposition of the following waves which will be given as -

y (x, t) = y₁ (x, t) + y₂ (x, t)

y (x, t) = 2 A sin { [(k₁ + k₂) / 2] x - [($\omega$₁ + $\omega$₂) / 2] t }

y (x, t) = [2 (2 m)] sin { [(3.349 m^-1) / 2] x - [(50.24 rad/s) / 2] t }

y (x, t) = (4 m) sin [(1.674 m^-1) - (25.1 rad/s) t }