Question

A source emit sound waves of frequency \(1000 \mathrm{~Hz}\). The source moves to the right with a speed of \(32 \mathrm{~m} / \mathrm{s}\) relative to ground. On the right a reflecting surface moves towards left with a speed of \(64 \mathrm{~m} / \mathrm{s}\) relative to ground. The speed of sound in air is \(332 \mathrm{~m} / \mathrm{s}\), then which of the following options is incorrect : (1) wavelength of sound in ahead of source is \(0.3 \mathrm{~m}\) (2) number of waves arriving per second which meets the reflected surface is 1320 (3) speed of reflected wave is \(268 \mathrm{~m} / \mathrm{s}\) (4) wavelength of reflected waves is nearly \(0.2 \mathrm{~m}\)

Short answer

Option (1) is incorrect.

Step-by-step solution

- Calculate the wavelength of the sound ahead of the source

The wavelength \( \lambda \) is given by the formula \[ \lambda = \frac{v_{sound}}{f} \]. Considering that the source is moving to the right, the effective speed of sound relative to the source is \(v_{sound} + v_{source} = 332 \, m/s + 32 \, m/s = 364 \, m/s\). So, the wavelength \lambda_{\text{ahead}} = \frac{364 \, m/s}{1000 \, Hz} = 0.364 \, m\.

- Determine the number of waves arriving per second at the reflecting surface

Using the Doppler effect, the frequency perceived by the reflecting surface can be calculated as \[ f' = \left( \frac{v_{sound} - v_{surface}}{v_{sound} - v_{source}} \right) f \], where \(v_{sound}\) = 332 \, m/s, \(v_{source}\) = 32 \, m/s, and \(v_{surface}\) = 64 \, m/s. Substituting the values, \[ f' = \left( \frac{332 \, m/s + 64 \, m/s}{332 \, m/s - 32 \, m/s} \right) \, 1000 \, Hz = \left( \frac{396}{300} \right) 1000 \, Hz = 1320 \, Hz \].

- Calculate the speed of the reflected wave

The speed of the reflected wave will be the same as the speed of sound traveling through the medium, i.e., \(332 \, m/s\). However, since both the source and surface are moving, the speed relative to the ground of the reflected wave is perceived differently. But it does not affect the intrinsic speed of sound which stays \(332 \, m/s\).

- Determine the wavelength of the reflected wave

The wavelength of the reflected wave can be found using \lambda_r = \frac{v_{sound}'}{f'}\, where \(v_{sound}'\) is the speed of the sound relative to the moving reflector which moves toward the source at \(68 \, m/s\) effectively. Their sum for the wavelength calculation is \[ \lambda_r = \frac{332 \, m/s}{1320 \, Hz} = 0.251 \, m \].

- Identify the incorrect statement

Compare all statements to the calculated results:(1) Wavelength ahead of source: 0.364 m (incorrect)(2) Number of waves arriving per second: 1320 (correct)(3) Speed of reflected wave: 332 m/s (correct)(4) Wavelength of reflected wave: 0.251 m (nearly correct, but less than 0.3 m)So, Option (1) is incorrect.

Key concepts

Sound Waves

Sound waves are vibrations that travel through a medium, such as air, water, or solids. They are created by the vibration of an object, which causes the surrounding medium to be compressed and rarefied, creating a wave. These waves carry energy and travel in a straight line outward from the source. When we hear a sound, it’s because these waves have reached our ears.

Sound waves can be categorized into the following types:

• Longitudinal Waves: In these waves, the displacement of the medium is parallel to the direction of the wave propagation. This is the type of wave that sound usually propagates as.
• Transverse Waves: In these waves, the displacement of the medium is perpendicular to the direction of the wave propagation. Although sound waves primarily propagate as longitudinal waves, they can exhibit transverse characteristics under certain conditions.

The speed of sound in air at room temperature is approximately 332 meters per second. This speed can change based on factors like temperature, pressure, and the medium through which it travels.

Frequency

Frequency refers to the number of wave cycles that pass a point per second, measured in Hertz (Hz). For sound waves, the frequency determines the pitch of the sound we hear. A higher frequency corresponds to a higher pitch, while a lower frequency corresponds to a lower pitch.

The Doppler effect is a phenomenon that affects the frequency of sound waves when there is relative motion between the source of the sound and the observer. If the source is moving towards the observer, the frequency appears to increase (the sound is higher-pitched), and if the source is moving away, the frequency appears to decrease (the sound is lower-pitched).

In the context of the solution, let's consider two scenarios:

• Source Moving Towards Observer: The frequency perceived increases due to the compression of sound waves.
• Source Moving Away from Observer: The frequency perceived decreases as the sound waves stretch out and their separation increases.

Wave Propagation

Wave propagation refers to the movement of waves through a medium. For sound waves, this means how the waves travel through air, water, or other substances. The process can be influenced by the properties of the medium as well as the relative motion between the source and the medium.

Waves can undergo different interactions during propagation, including:

• Reflection: When waves hit a surface and bounce back. This is the principle behind echoes.
• Refraction: When waves pass into a medium with different properties, causing them to change direction.
• Diffraction: When waves bend around obstacles or through slits.
• Interference: When waves overlap, they can add together or cancel each other out.

In our exercise, we see reflection in action when the sound hits a moving surface and bounces back. The relative motion of the surfaces alters the wave properties, like frequency and wavelength. Understanding these interactions is crucial for applications ranging from acoustics to communication technologies.