Question 416: Waves

Question

What is the effect of pressure on the speed of sound in a medium, if pressure is doubled at constant temperature?

Accepted Answer

The speed of sound in a medium is influenced by the properties of the medium itself. Specifically, it depends on the medium's density and elasticity. The formula for the speed of sound $$v$$ in a medium is given by:
$$ v = \sqrt{\frac{E}{\rho}} $$
Here, $$E$$ represents the modulus of elasticity (such as Bulk modulus for gases, Young's modulus for solids), and $$\rho$$ represents the density of the medium.
For gases, we can further refine this relationship using the ideal gas law. For example, in an ideal gas, the speed of sound can be expressed as:
$$ v = \sqrt{\frac{\gamma P}{\rho}} $$
where:
$$\gamma$$ is the adiabatic index (ratio of specific heats $$C_p/C_v$$).
$$P$$ is the pressure of the gas.
$$\rho$$ is the density of the gas.
At constant temperature, the density ($$\rho$$) of a gas is directly proportional to its pressure ($$P$$) according to the ideal gas law:
$$ P \propto \rho \implies \rho \propto P $$
Thus, if pressure $$P$$ is doubled (keeping temperature constant), the density $$\rho$$ will also double. Substituting this into the speed of sound equation gives:
$$ v = \sqrt{\frac{\gamma P}{\rho}} $$
If $$P$$ is doubled and $$\rho$$ is also doubled, the effect on the speed of sound is:
$$ v = \sqrt{\frac{\gamma (2P)}{2\rho}} = \sqrt{\frac{2\gamma P}{2\rho}} = \sqrt{\frac{\gamma P}{\rho}} $$
We see that the expression for the speed of sound remains unchanged. Therefore, doubling the pressure at constant temperature does not affect the speed of sound in the medium.
Therefore, the correct option is:
Option A: Remains same

Answer

Remains same

Answer

Reduced to half

Answer

Gets doubled

Answer

Becomes 4 times

Waves

What is the effect of pressure on the speed of sound in a medium, if pressure is doubled at constant temperature?

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