Q.Change in temperature of the medium changes

The speed of sound in a gas depends on the temperature of the medium.

1. Concept: Speed of Sound in Gases

   The speed of sound $v$ in a gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$, where $\gamma$ is the adiabatic index, $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass. This shows that $v \propto \sqrt{T}$. Therefore, a change in temperature directly changes the speed of sound.

2. The frequency $f$ of a sound wave is determined by its source and remains constant regardless of the medium's properties or temperature changes. …

The speed at which sound travels through a medium is a property of the medium itself. For gases, this speed is significantly influenced by temperature. When the speed of a wave changes, but its frequency (which is determined by the source) remains constant, its wavelength must adjust to maintain the fundamental wave relationship.

Let's break down how temperature affects the different properties of a sound wave.

1. Speed of Sound and Temperature

   The speed of sound in an ideal gas is given by the formula:

   $v = \sqrt{\frac{\gamma RT}{M}}$

   where:

   • $v$ is the speed of sound
   • $\gamma$ is the adiabatic index (ratio of specific heats, a constant for a given gas)
   • $R$ is the universal gas constant
   • $T$ is the absolute temperature of the gas (in Kelvin)
   • $M$ is the molar mass of the gas

   From this formula, it is clear that the speed of sound $v$ is directly proportional to the square root of the absolute temperature $T$ ($v \propto \sqrt{T}$). This means that if the temperature of the medium changes, the speed of sound in that medium will also change. Specifically, an increase in temperature increases the speed of sound, and a decrease in temperature decreases it.

2. Frequency of Sound Waves

   The frequency of a sound wave is determined solely by the source that produces it. For example, if a speaker vibrates at 500 Hz, it will generate sound waves with a frequency of 500 Hz, regardless of the air temperature or any other property of the medium. The frequency represents the rate at which the source oscillates, and this rate is intrinsic to the source itself. Therefore, a change in the temperature of the medium does not change the frequency of the sound waves.

3. Wavelength of Sound Waves

   The fundamental relationship between the speed of a wave ($v$), its frequency ($f$), and its wavelength ($\lambda$) is given by the wave equation:

We can rearrange this to express wavelength:

From Step 1, we established that the speed of sound ($v$) changes with temperature. From Step 2, we know that the frequency ($f$) remains constant because it is determined by the source. …

Step 1: $v=\sqrt{\gamma RT/M}$ shows the speed of sound depends on temperature ($v\propto\sqrt T$); the frequency is fixed by the vibrating source and doesn't depend on the medium's temperature.

Step 2: Since $v=f\lambda$ and $f$ stays fixed while $v$ changes with $T$, the wavelength $\lambda=v/f$ must change to compensate. …