### Standing Waves In A Pipe Phenomena Of Beats And Doppler Effect

**Standing waves in a pipe**

**Pressure variation in a pipe**

- The pressure variation in a pipe is represented by the equation ( P = P_0\cos(kx-\omega t)), where (P_0) is the amplitude of the pressure variation, (k) is the wave number, (x) is the position along the pipe, and (t) is the time.

**Equation of pressure variation in a pipe**

- (P = P_0\cos(kx - \omega t))

**Pressure nodes and displacement antinodes**

- Pressure nodes are points where the pressure variation is zero, while displacement antinodes are points where the displacement of the medium is maximum.
- The pressure nodes occur at the ends of the pipe, while the displacement antinodes occur in the middle of the pipe.

**Displacement nodes and pressure antinodes**

- Displacement nodes are points where the displacement of the medium is zero, while pressure antinodes are points where the pressure variation is maximum.
- The displacement nodes occur in the middle of the pipe, while the pressure antinodes occur at the ends of the pipe.

**Harmonics and their frequencies**

- Harmonics are multiples of the fundamental frequency of the standing wave.
- The frequencies of the harmonics are given by ( f_n = nf_1 ), where (f_n) is the frequency of the nth harmonic, (f_1) is the fundamental frequency, and (n) is a positive integer.

## **Phenomena of beats** ---

**Definition of beats**

- Beats are the periodic variations in the amplitude of a sound wave caused by the interference of two waves with slightly different frequencies.

**Production of beats**

- Beats are produced when two sound waves with slightly different frequencies are combined. The waves interfere with each other, resulting in variations in the amplitude of the sound.

**Equation of beat frequency**

- The beat frequency is given by ( f_b = |f_1-f_2|), where (f_b) is the beat frequency, (f_1) is the frequency of the first wave, and (f_2) is the frequency of the second wave.

**Determination of beat frequency**

- The beat frequency can be determined by listening to the sound waves and counting the number of beats per second.

**Applications of beats**

- Beats are used in music to create a variety of effects, such as vibrato and tremolo.
- Beats are also used in physics to measure the frequency of sound waves.

## **Doppler effect** --- #### **Definition of Doppler effect**

- The Doppler effect is the change in frequency of a wave due to the relative motion of the source and the observer.

**Types of Doppler effect: approaching and receding sources**

- The Doppler effect can be divided into two types: the approaching source and the receding source.
- The approaching source occurs when the source is moving towards the observer, while the receding source occurs when the source is moving away from the observer.

**Equation for the Doppler effect**

- The equation for the Doppler effect is given by ( f_o = f_s\frac{(v+v_0)}{v \pm v_s}), where ( f_o) is the observed frequency, ( f_s) is the source frequency, (v) is the speed of the wave, ( v_0) is the speed of the observer, and ( v_s ) is the speed of the source.
- The positive sign in the denominator is used for the approaching source, while the negative sign is used for the receding source.

**Determination of Doppler effect**

- The Doppler effect can be determined by measuring the change in frequency of the wave.

**Applications of the Doppler effect**

- The Doppler effect is used in a variety of applications, such as radar, sonar, and astronomy.