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 Acoustic Plane Wave Properties
  Acoustic Pressure
  Acoustic Pressure Addition

Acoustic Plane Wave Properties

The acoustic pressure caused by an acoustic disturbance varies with both time and position. Besides studying of the fluctuation of acoustic pressure that associated with a sound wave with respective to time at any point, many practical problems focus on the pressure variation caused by the sound at any point with respective to position.

Acoustic Pressure

The acoustic pressure at any point varies with time sinusoidally, the root mean square is more convenient to be used to measure the alternating pressure fluctuation by taking the square root of the average of squares of instantaneous acoustic pressures within a period T. That is

image

where

prms is the root mean square of acoustic pressures
T is the period of measurement
p is the instantaneous acoustic pressures

The instantaneous acoustic pressure can be expressed as:

image

where

p(x,t) is the instantaneous acoustic pressures
x is distance
t is time
pm is the maximum of the instantaneous acoustic pressures
ω is angular velocity
k is wave number
ƒ is frequency
λ is wave length

Therefore:

  image

Since the time average of a cosine function of function of time equals to zero, imply :

image or image

Acoustic Pressure Addition

The addition of two sound pressures of different frequency equals to:

image

Imply the square of the total sound pressure is

  image

The root mean square value of the total the total acoustic pressure can be obtained by taking the time average process. Similarly, the cosine term of function of time equal s to zero, and get

image

Therefore. the total acoustic pressure of the two equal sound pressure due to two noises of different frequencies are only the √2 times of the sound pressure due to either noise source. The result can also be extended to a continuous bands of frequencies. However, when the two frequencies are equal and coherent, the result will be different.

When there is the present of phase angle, the result becomes

image

Since the time average of the cosine terms of functions of time is zero, the phase angle does not affect the result of sounds with different frequencies. However, the phase angle affects the result when the frequencies of two sound sources are the same. By taking the time average, the equation with same frequency becomes:

image

Acoustic Intensity

Acoustic intensity is the sound power passing through a unit area, it can be considered as the rate of transmitting energy through the medium cross section area by medium fluctuation due to sound wave. The vibration of medium composes the change in both potential and kinetic energy.

Assuming constant mass and at initial state both potential energy and kinetic energy are zero with the acoustic disturbance equals zero. The potential energy EPE and kinetic energy EKE of an infinitesimal element of volume Vo due to acoustic fluctuation along x direction is:

imageimage

From the linearized continuity equation, where s is the acoustic displacement,

image

Substitute the speed of propagation,

image

The volume of the infinitesimal element can be expressed as,

image

Substitute the differential in term of pressure, then

image

The differential volume can then be equals to:

image

Therefore the potential energy is:

image

For the kinetic energy, substitute the velocity in term of pressure :

image

Therefore the total instantaneous energy of an infinitesimal element due to acoustic disturbance is:

image

Since the wave propagate at the speed of s, the volume of infinitesimal element per unit area in time δt  can be expressed as:

image

As the instantaneous intensity is the rate of energy flow through a unit area, imply:

image

Or, in term of acoustic pressure and acoustic velocity:

image

Similar to the pressure, the intensity usually expressed as an average in practice, imply:

image

In form of complex exponential representation, the intensity equals to the real active component of intensity:

image

Imply

image

Acoustic Power of Plane Wave

Sound power is defined as the integration of intensity over the surface area enclosing the enclosure and is

image

SPL and SIL of Plane Wave

The Sound Pressure Level of a plane wave:

image

Assume ρo = 1.2kg/m3, c = 334m/s,the Sound Intensity Level of a plane wave:

image

Plane Wave Example

A plane wave propagates with a frequency of 50Hz and maximum pressure amplitude 150Pa in positive x direction with characteristic that at time t = 0 and at x = 0, the plane wave in term of acoustic pressure can be expressed as:

image

The corresponding plane wave in term of medium velocity can be expressed as:

image

The mean intensity of the plane wave can be expressed as:

image

The sound intensity level of the plane wave in term of dB can be expressed as:

image

©sideway

ID: 100900023 Last Updated: 9/10/2010 Revision: 1 Ref:

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References

  1. Michael P. Norton; Denis G. Karczub,, 2003
  2. G. Porges, 1977
  3. Douglas D. Reynolds, 1981
  4. Conrad J. Hemond, 1983
  5. F. Fahy, 2001
  6. D.A. Biew; C.H. Hansen, 1996
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