Standing Waves In A Pipe

Resonance Resonance occurs in a pipe when the length of the pipe is an integer multiple of half the wavelength of the sound wave. This condition can be expressed mathematically as:

$$L = n\frac{\lambda}{2}$$

Where:

• $$L$$ is the length of the pipe
• $$\lambda$$ is the wavelength of the sound wave
• $$n$$ is an integer

When resonance occurs, the sound wave reflects back and forth between the ends of the pipe, creating a standing wave. The frequency of the standing wave is given by:

$$f = n\frac{v}{2L}$$

Where:

• $$f$$ is the frequency of the standing wave
• $$v$$ is the speed of sound in the air
• $$L$$ is the length of the pipe
• $$n$$ is an integer

Nodes and Antinodes Nodes are points in a standing wave where the amplitude of the wave is zero. Antinodes are points in a standing wave where the amplitude of the wave is maximum. The positions of the nodes and antinodes in a pipe are given by:

$$x_n = n\frac{\lambda}{2}$$

Where:

• $$x_n$$ is the position of the nth node or antinode
• $$\lambda$$ is the wavelength of the sound wave
• $$n$$ is an integer

For more information, refer to NCERT Class 11 Physics, Chapter 15: Waves

Phenomena Of Beats

Beats are the result of the interference of two sound waves with slightly different frequencies. When two sound waves of different frequencies are superimposed, the resultant wave oscillates with a frequency that is equal to the average of the two frequencies. However, the amplitude of the resultant wave varies periodically, creating the illusion of a beating sound.

The beat frequency, $$f_b$$, is given by:

$$f_b = |f_1 - f_2|$$

Where:

• $$f_1$$ is the frequency of the first sound wave
• $$f_2$$ is the frequency of the second sound wave

Beats can be heard as a warbling or pulsating sound. The phenomenon of beats is used in tuning musical instruments.

For more information, refer to NCERT Class 11 Physics, Chapter 15: Waves

Doppler Effect

The Doppler effect is the change in frequency of a sound wave due to the motion of the source or the observer. When the source of the sound wave is moving towards the observer, the frequency of the wave will be higher than if the source were stationary. This is because the sound waves are compressed as they approach the observer, resulting in a higher frequency. Conversely, when the source of the sound wave is moving away from the observer, the frequency of the wave will be lower than if the source were stationary. This is because the sound waves are stretched out as they recede from the observer, resulting in a lower frequency.

The amount of the frequency shift, $$\Delta f$$, depends on the speed of the source, $$v_s$$, the speed of sound in the air, $$v$$, and the angle, $$\theta$$, between the direction of motion of the source and the line connecting the source and the observer. The Doppler effect is given by the equation:

$$\Delta f = \pm\frac{v_s}{v}\cos\theta$$

Where:

• $$\Delta f$$ is the change in frequency
• $$v_s$$ is the speed of the source
• $$v$$ is the speed of sound in the air
• $$\theta$$ is the angle between the direction of motion of the source and the line connecting the source and the observer

The positive sign is used when the source is moving towards the observer, and the negative sign is used when the source is moving away from the observer.

For more information, refer to NCERT Class 11 Physics, Chapter 15: Waves