Two transverse waves of the same angular frequency travel along a string in the same…

PHYSICS · Multi-part · A Level · Physics · Waves · Waves

Problem

Two transverse waves of the same angular frequency

\omega

travel along a string in the same direction. Their displacements are given by

y_1 = A_1 \sin \omega t, \qquad y_2 = A_2 \sin (\omega t + \theta)

, where

A_1 \neq A_2

. (i) Derive an expression for the amplitude of the resultant wave in terms of

A_1

A_2

, and

\theta

. (ii) Obtain an expression for the phase of the resultant wave with respect to the first wave. (iii) Calculate the resultant amplitude when

A_1 = 4.0\,\text{cm}

A_2 = 3.0\,\text{cm}

, and

\theta = 120^{\circ}

. (iv) For these amplitudes, find the value(s) of

\theta

(if any) that give complete cancellation at the point. Justify mathematically.