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A sound wave propagates in a medium of B...

A sound wave propagates in a medium of Bulk modulus B by means of compressions and rare fractions. If `P_( c)` and `P_(e )` are the pressures at compression and rarefaction respectively, .a. be the wave amplitude and k be the angular wave number then,

A

`P_c` is maximum and `P_r` is minimum.

B

`P_c` is minimum and `P_r` is maximum

C

the pressure amplitude is Bak

D

if the displacement wave is `Y = a sin(omegat -kx)` , the pressure wave at an instant is represented as `(P_(c)-P_(r))/2 cos(omegat -kx)` Which leads displacement wave by a phase angle of `pi//2`.

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The correct Answer is:
To solve the problem regarding the sound wave propagation in a medium with a given bulk modulus, we need to analyze the properties of sound waves, particularly focusing on pressure variations during compressions and rarefactions. ### Step-by-Step Solution: 1. **Understanding Pressure at Compression and Rarefaction**: - In a sound wave, compressions occur where the pressure is higher than the average atmospheric pressure, while rarefactions occur where the pressure is lower. - Therefore, we can conclude that at points of compression (denoted as \( P_c \)), the pressure is maximum, and at points of rarefaction (denoted as \( P_r \)), the pressure is minimum. **Hint**: Remember that compressions correspond to areas of high pressure, while rarefactions correspond to areas of low pressure. 2. **Identifying the Correct Options**: - From our understanding, the first option states that \( P_c \) is maximum and \( P_r \) is minimum. This is correct. - The second option states that \( P_c \) is minimum and \( P_r \) is maximum, which is incorrect. **Hint**: Compare the definitions of compressions and rarefactions to determine the correct relationship between \( P_c \) and \( P_r \). 3. **Pressure Amplitude Expression**: - The pressure amplitude in a sound wave can be expressed as \( \Delta P = B \cdot a \cdot k \), where \( B \) is the bulk modulus, \( a \) is the wave amplitude, and \( k \) is the angular wave number. - Thus, the third option stating that the pressure amplitude is \( B \cdot a \cdot k \) is correct. **Hint**: Recall the relationship between pressure amplitude and the properties of the wave, including the bulk modulus and wave amplitude. 4. **Phase Relationship Between Pressure and Displacement Waves**: - The displacement wave can be represented as \( y = a \sin(\omega t - kx) \). - The pressure wave, due to the nature of sound waves, leads the displacement wave by a phase angle of \( \frac{\pi}{2} \). - Therefore, the pressure wave can be expressed as \( P = P_c - P_r \) and leads the displacement wave by \( \frac{\pi}{2} \). **Hint**: Consider the phase difference in wave mechanics, especially how pressure and displacement waves relate in terms of their sinusoidal representations. 5. **Conclusion**: - Based on the analysis, the correct options are: - Option 1: \( P_c \) is maximum and \( P_r \) is minimum. - Option 3: Pressure amplitude is \( B \cdot a \cdot k \). - Option 4: The pressure wave leads the displacement wave by \( \frac{\pi}{2} \). - The incorrect option is Option 2. ### Final Answer: - Correct Options: 1, 3, and 4. - Incorrect Option: 2.
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