(A) The phase difference between the two waves is given by $\phi = (\omega t - \beta_2) - (\omega t - \beta_1) = \beta_1 - \beta_2$.According to the p

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$\sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\beta_1 - \beta_2)}$

(A) The phase difference between the two waves is given by $\phi = (\omega t - \beta_2) - (\omega t - \beta_1) = \beta_1 - \beta_2$.According to the principle of superposition,the resultant amplitude $A$ of two waves with amplitudes $A_1$ and $A_2$ and phase difference $\phi$ is given by the formula $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi}$.Substituting the value of $\phi$,we get $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\beta_1 - \beta_2)}$.

Class 11 Physics Waves and Sound Principle of superposition of waves

Easy

Two waves $y_1 = A_1 \sin(\omega t - \beta_1)$ and $y_2 = A_2 \sin(\omega t - \beta_2)$ superimpose to form a resultant wave whose amplitude is:

A
$\sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\beta_1 - \beta_2)}$
B
$\sqrt{A_1^2 + A_2^2 + 2A_1A_2 \sin(\beta_1 - \beta_2)}$
C
$A_1 + A_2$
D
$|A_1 + A_2|$

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Waves and Sound

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Principle of superposition of waves

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.

Difficult

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The displacement equations of two interfering waves are given by
$y_1 = 10 \sin \left(\omega t + \frac{\pi}{3}\right) \text{ cm}$
$y_2 = 5[\sin (\omega t) + \sqrt{3} \cos \omega t] \text{ cm}$ respectively.
The amplitude of the resultant wave is $............. \text{ cm}$.

DifficultJEE MAIN 2023

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Three waves of equal frequency having amplitudes $10 \,\mu m, 4 \,\mu m$ and $7 \,\mu m$ arrive at a given point with successive phase difference of $\frac{\pi}{2}$. The amplitude of the resulting wave in $\mu m$ is given by

MediumAIIMS 1995

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Two pulses in a stretched string whose centres are initially $8 \ cm$ apart are moving towards each other as shown in the figure. The speed of each pulse is $2 \ cm/s$. After $2 \ seconds$,the total energy of the pulses will be

MediumIIT 2001

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Two sinusoidal waves given below are superposed:
$y_1 = A \sin \left(kx - \omega t + \frac{\pi}{6}\right), \quad y_2 = A \sin \left(kx - \omega t - \frac{\pi}{6}\right)$
The equation of the resultant wave is:

Medium

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Two waves $y_1 = A_1 \sin(\omega t - \beta_1)$ and $y_2 = A_2 \sin(\omega t - \beta_2)$ superimpose to form a resultant wave whose amplitude is:

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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.

The displacement equations of two interfering waves are given by$y_1 = 10 \sin \left(\omega t + \frac{\pi}{3}\right) \text{ cm}$$y_2 = 5[\sin (\omega t) + \sqrt{3} \cos \omega t] \text{ cm}$ respectively.The amplitude of the resultant wave is $............. \text{ cm}$.

Three waves of equal frequency having amplitudes $10 \,\mu m, 4 \,\mu m$ and $7 \,\mu m$ arrive at a given point with successive phase difference of $\frac{\pi}{2}$. The amplitude of the resulting wave in $\mu m$ is given by

Two pulses in a stretched string whose centres are initially $8 \ cm$ apart are moving towards each other as shown in the figure. The speed of each pulse is $2 \ cm/s$. After $2 \ seconds$,the total energy of the pulses will be

Two sinusoidal waves given below are superposed:$y_1 = A \sin \left(kx - \omega t + \frac{\pi}{6}\right), \quad y_2 = A \sin \left(kx - \omega t - \frac{\pi}{6}\right)$The equation of the resultant wave is:

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The displacement equations of two interfering waves are given by
$y_1 = 10 \sin \left(\omega t + \frac{\pi}{3}\right) \text{ cm}$
$y_2 = 5[\sin (\omega t) + \sqrt{3} \cos \omega t] \text{ cm}$ respectively.
The amplitude of the resultant wave is $............. \text{ cm}$.

Two sinusoidal waves given below are superposed:
$y_1 = A \sin \left(kx - \omega t + \frac{\pi}{6}\right), \quad y_2 = A \sin \left(kx - \omega t - \frac{\pi}{6}\right)$
The equation of the resultant wave is: