The Laplace's correction in the expression for the velocity of sound given by Newton is needed because sound waves

Options

A. are longitudinal
B. propagate isothermally
C. progagate adiabatically
D. are of long wavelength

Answer

To solve the question regarding the need for Laplace's correction in the expression for the velocity of sound, we can break it down into the following steps:

### Step 1: Understand the Original Expression for Velocity of Sound
The original expression for the velocity of sound given by Newton is:
\[ v = \sqrt{\frac{p}{\rho}} \]
where \( p \) is the pressure and \( \rho \) is the density of the medium (in this case, air).

### Step 2: Identify the Issue with Newton's Expression
When we apply Newton's formula under normal atmospheric conditions, we find that the calculated velocity of sound is approximately 280 m/s. However, experimental values show that the actual velocity of sound in air is around 332 m/s. This discrepancy indicates that Newton's expression is not entirely accurate.

### Step 3: Introduce the Laplace Correction
To correct the discrepancy, Laplace introduced a correction factor, denoted as \( \gamma \) (gamma), which accounts for the adiabatic nature of sound wave propagation. The corrected formula for the velocity of sound becomes:
\[ v = \sqrt{\gamma \frac{p}{\rho}} \]

### Step 4: Understand the Role of \( \gamma \)
The factor \( \gamma \) is the ratio of specific heats (Cp/Cv) of the gas, which is important because sound waves in gases propagate as longitudinal waves involving alternating compressions and rarefactions. This means that the temperature of the gas changes during the propagation of sound, and thus the process is adiabatic (no heat exchange with the surroundings).

### Step 5: Final Expression for Velocity of Sound
The corrected expression can also be represented in terms of temperature and molecular weight:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molecular weight of the gas.

### Conclusion
The Laplace correction is necessary because sound waves propagate adiabatically, causing periodic changes in temperature, which Newton's original formula did not account for. Therefore, the correct answer is that sound waves propagate adiabatically, leading to the need for the Laplace correction.

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