Tru Physics

Longitudinal Waves

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Introduction

Longitudinal waves are a type of wave in which the displacement of the medium’s particles is parallel to the propagation of the wave. They are characterized by alternating regions of compressions (areas where the particles are close together) and rarefactions (areas where the particles are far apart).

Wave Parameters and Properties

Wave Speed

The speed of a wave is given by the formula:

v = \lambda f

where v is the wave speed, \lambda is the wavelength, and f is the frequency of the wave.

Wave Function

The wave function for a one-dimensional longitudinal wave can be expressed as:

y(x, t) = A \cos(kx - \omega t + \phi)

where y is the displacement of the particle from its equilibrium position, A is the amplitude of the wave, k is the wave number, x is the position, \omega is the angular frequency, t is time, and \phi is the phase constant.

Examples of Longitudinal Waves

Sound Waves

Sound waves are an example of longitudinal waves. They propagate through the air (or another medium) by compressing and expanding the medium. The speed of sound depends on the medium and the temperature, but in air at room temperature, it is approximately 343 m/s.

Seismic P-Waves

Seismic primary waves, or P-waves, are also longitudinal waves. They are the fastest seismic waves and the first to be detected by seismographs in an earthquake.

Wave Interference

When two or more waves overlap, they interfere with each other. The principle of superposition states that the total displacement of the medium at any point is the sum of the displacements due to each individual wave. For two waves with the same amplitude, frequency, and wavelength:

y_{\text{total}}(x, t) = y_1(x, t) + y_2(x, t)

where y_{\text{total}} is the total displacement, and y_1 and y_2 are the displacements due to each wave.

Standing Waves and Resonance

When two waves of the same frequency, amplitude, and wavelength travel in opposite directions and interfere, they can form a standing wave. At certain points, known as nodes, there is no movement, while at others, known as antinodes, the displacement is maximum.

The phenomenon of resonance occurs when an object is forced to vibrate at its natural frequency, leading to a significant increase in amplitude. This is fundamental in musical instruments and many other systems.

Doppler Effect

The Doppler effect refers to the change in frequency or wavelength of a wave for an observer moving relative to the source of the wave. The frequency increases (and wavelength decreases) if the observer and source are moving towards each other and decreases (and wavelength increases) if they are moving apart.

For sound waves, the observed frequency f' is given by:

f' = \dfrac{(v + v_0)}{(v + v_s)}f

where v is the speed of sound, v_0 is the speed of the observer (positive if moving towards the source), v_s is the speed of the source (positive if moving away from the observer), and f is the emitted frequency.

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