Two waves represented by y = a sin(ωt - kx) and y = a cos(ωt - kx)

1.	Two waves represented by y = a sin(ωt - kx) and y = a cos(ωt - kx) are superposed. The resultant wave will have an amplitude(a) a(b)√2a(c) 2a(d) 0.
Answer» The correct answer is (b)√2a EXPLANATION: When two waves are superimposed, the displacements of the particle due to each wave is added. Hence the resultant wave is given as y' = a sin(⍵t-kx) + a cos(⍵t-kx) →y' = a {sin(⍵t-kx) + sin(π/2+⍵t-kx)} →y'=a2 sin{(⍵t-kx+π/2+⍵t-kx)/2}cos{(⍵t-kx-π/2-⍵t+kx)/2} →y' = 2a sin(⍵t-kx+π/4)*cosπ/4 →y' = (2a/√2) sin(⍵t-kx+π/4) →y' =√2a sin(⍵t-kx+π/4) Clearly the amplitude of the resultant wave is √2a.

Two waves represented by y = a sin(ωt - kx) and y = a cos(ωt - kx) are superposed. The resultant wave will have an amplitude(a) a(b)√2a(c) 2a(d) 0.

Answer»

The correct answer is (b)√2a

EXPLANATION:

When two waves are superimposed, the displacements of the particle due to each wave is added. Hence the resultant wave is given as

y' = a sin(⍵t-kx) + a cos(⍵t-kx)

→y' = a {sin(⍵t-kx) + sin(π/2+⍵t-kx)}

→y'=a*2 sin{(⍵t-kx+π/2+⍵t-kx)/2}*cos{(⍵t-kx-π/2-⍵t+kx)/2}

→y' = 2a sin(⍵t-kx+π/4)*cosπ/4

→y' = (2a/√2) sin(⍵t-kx+π/4)

→y' =√2a sin(⍵t-kx+π/4)

Clearly the amplitude of the resultant wave is √2a.