Question

Two interfering waves have the same wavelength, frequency, and amplitude. They are traveling in the same direction but are ${90}^o$ out of phase. Compared to the individual waves, the resultant wave will have the same.

• A. amplitude and velocity but different wavelength
• B. amplitude and wavelength but different velocity
• C. wavelength and velocity but different amplitude
• D. amplitude and frequency but different velocity

Answer 1

The correct option is C wavelength and velocity but different amplitude

let the equation of wave be $Asin(kx-\omega t)$ and $Asin(kx-\omega t+\frac{\pi }{2})$ which can also be written as $Acos(kx-\omega t)$

so the resultant is $S=Asin(kx-\omega t)+Acos(kx-\omega t)$

by observing both the waves on the graph we can see that the shape of the wave remain the same but with higher amplitude

$S=\sqrt{A^{2}si{n}^2(kx-\omega t)+A^{2}co{s}^2(kx-\omega t)+2A^{2}sin(kx-\omega t)cos(kx-\omega t)}$

$S=\sqrt{A^2+2A^{2}sin(kx-\omega t)cos(kx-\omega t)}$

$sin2A=2sinAcosA$ and $sin2A=1-2si{n}^{2}A$

$S=A\sqrt{1+sin(2kx-2\omega t)}=A\sqrt{1+1-2si{n}^2(kx-\omega t)}=\sqrt{2}Acos(kx-\omega t)$