Suppose that two waves travel simultaneously along the same stretched string. Let $1($x$,$ t$)$ and yz$($x$,$ t$)$ be the displacements that the string would experience if each wave traveled alone. The displacement of the string when the waves overlap is then the algebraic sum

y $($x$,$ t$)=41($x$,$ t$)+$ Y$2($x$,$ t$).$

Let one wave traveling along a stretched string be given by:

V$1($x$,$ t$)=$ Ymsin$($kx $-$ wt$),$ and another by Yz$($x$,$ t$)=$ Ymsin$($kx $-$ wt $+).$

$$Using the principle of superposition, derive an expression for the algebraic sum of the two interfering waves.

$[$Useful Trigonometric Relation: sin a $+$ sin B $=2$ sin$(*+$B$)$ cos $(*=)]$

$$The above expression for y$1($x$,$ t$)+$ Y$2($x$,$ t$)$ can be expressed as a single wave in the form y $($x$,$ t$)=$ A sin$($kx $-$ wt $+\text{'}).$ Identify the resultant amplitude A and phase constant d$\text{'}$ of the superimposed wave y $($x$,$ t$).$

$$Interference that produces the greatest possible amplitude is called fully constructive interference. What is the value of d if the waves described in earlier parts interfere fully constructively?

$$Interference that produces zero amplitude is called fully destructive interference. What is the value of o if the waves described in earlier parts interfere fully destructively?