Wave Equation in Higher Dimensions
The wave equation in 1D, 2D, or 3D may be written as
where, in 3D, denotes the amplitude of the wave at time and position , and
Plane Waves in Air
Figure B.9 shows a 2D cross-section of a snapshot (in time) of the sinusoidal plane wave
Figure B.10 depicts a more mathematical schematic of a sinusoidal plane wave traveling toward the upper-right of the figure. The dotted lines indicate the crests (peak amplitude location) along the wave.
The direction of travel and spatial frequency are indicated by the vector wavenumber , as discussed in in the following section.
Vector Wavenumber
Mathematically, a sinusoidal plane wave, as in Fig.B.9 or Fig.B.10, can be written as
where p(t,x) is the pressure at time (seconds) and position (3D Euclidean space). The amplitude , phase , and radian frequency are ordinary sinusoid parameters [451], and is the vector wavenumber:
- (unit) vector of direction cosines
- (scalar) wavenumber along travel direction
- wavenumber along the travel direction in its magnitude
- travel direction in its orientation
To see that the vector wavenumber has the claimed properties, consider that the orthogonal projection of any vector onto a vector collinear with is given by [451].^{B.35}Thus, is the component of lying along the direction of wave propagation indicated by . The norm of this component is , since is unit-norm by construction. More generally, is the signed length (in meters) of the component of along . This length times wavenumber gives the spatial phase advance along the wave, or, .
For another point of view, consider the plane wave , which is the varying portion of the general plane-wave of Eq.(B.48) at time , with unit amplitude and zero phase . The spatial phase of this plane wave is given by
As we know from elementary vector calculus, the direction of maximum phase advance is given by the gradient of the phase :
Since the wavenumber is the spatial frequency (in radians per meter) along the direction of travel, we should be able to compute it as the directional derivative of along , i.e.,
Scattering of plane waves is discussed in §C.8.1.
Solving the 2D Wave Equation
Since solving the wave equation in 2D has all the essential features of the 3D case, we will look at the 2D case in this section.
Specializing Eq.(B.47) to 2D, the 2D wave equation may be written as
The 2D wave equation is obeyed by traveling sinusoidal plane waves having any amplitude , radian frequency , phase , and direction :
The sum of two such waves traveling in opposite directions with the same amplitude and frequency produces a standing wave. For example, if the waves are traveling parallel to the axis, we have
which is a standing wave along .
2D Boundary Conditions
We often wish to find solutions of the 2D wave equation that obey certain known boundary conditions. An example is transverse waves on an ideal elastic membrane, rigidly clamped on its boundary to form a rectangle with dimensions meters.
Similar to the derivation of Eq.(B.49), we can subtract the second sinusoidal traveling wave from the first to yield
Note that we can also use products of horizontal and vertical standing waves
To build solutions to the wave equation that obey all of the boundary conditions, we can form linear combinations of the above standing-wave products having zero displacement (``nodes'') along all four boundary lines:
where
The Wikipedia page (as of 1/31/10) on the Helmholtz equation provides a nice ``entry point'' on the above topics and further information.
3D Sound
The mathematics of 3D sound is quite elementary, as we will see below. The hard part of the theory of practical systems typically lies in the mathematical approximation to the ideal case. Examples include Ambisonics [158] and wave field synthesis [49].
Consider a point source at position . Then the acoustic complex amplitude at position is given by
The fundamental approximation problem in 3D sound is to approximate the complex acoustic field at one or more listening points using a finite set of loudspeakers, which are often modeled as a point source for each speaker.
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The Ideal Vibrating String
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Properties of Gases