7.1 When there are two traveling waves of the same wavelength and frequency (hen

Question

7.1 When there are two traveling waves of the same wavelength and frequency (hence the same velocity) in phase: fA (x, t) A sin(kx- ot) fs (x, t) B sin(kx ot) then it's clear that the actual wave you observe is fa B fA (x, t)+ fs (x, t (A B) sin(kx-ot) due to superposition principle. Namely, you observe the same wave form, except now the amplitude is A+ B. Now consider there are two waves of the same wavelength and frequency (hence the same velocity), but are not in phase: fA(x, t A sin(kx-ot) fa (x, t)-B sin(kx- t + ?) where p is a phase constant. Follow the steps prescribed below and derive the actually observed wave.

Explanation / Answer

fA(x,t)=A*sin(k*x-w*t)

fB(x,t)=B*sin(k*x-w*t+phi)

sum=A*sin(k*x-w*t) + B*sin(k*x-w*t+phi)

=A*sin(k*x-w*t) + B*sin(k*x-w*t) *cos(phi)+B*cos(k*x-w*t)*sin(phi)

=sin(k*x-w*t)*(A+B*cos(phi))+B*cos(k*x-w*t)*sin(phi)

let C=A+B*cos(phi)

D=B*sin(phi)

and theta=k*x-w*t

then sum=C*sin(theta)+D*cos(theta)

sum=sqrt(C^2+D^2)*((C/sqrt(C^2+D^2))*sin(theta)+(D/sqrt(C^2+D^2))*cos(theta))

let beta be angle such that

cos(beta)=C/sqrt(C^2+D^2)

and sin(beta)=D/sqrt(C^2+D^2)

sum=sqrt(C^2+D^2)*(cos(beta)*sin(theta)+sin(beta)*cos(theta))

=sqrt(C^2+D^2)*sin(theta+beta)

where C=A+B*cos(phi)

D=B*sin(phi)

theta=k*x-w*t

Q4.

part a:

amplitude of the wave observed=sqrt(C^2+D^2)

where C=A+B*cos(phi)

D=B*sin(phi)

part b:

speed of the wave=w/k

part c:

as seen from the final form,

sqrt(C^2+D^2)*sin(theta+beta)

the observed wave form still sinusoidal.

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