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.

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.

• 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.