“Wave Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Wave Equation

释义

Wave Equation

The wave equation is

(1)

where

is the Laplacian.

The 1-D wave equation is

(2)

In order to specify a wave, the equation is subject to boundary conditions

(3)

(4)

and initial conditions

			(5)
			(6)

The wave equation can be solved using the so-called d'Alembert's solution, a Fourier Transform method, orSeparation of Variables.

d'Alembert devised his solution in 1746, and Euler subsequently expanded the method in 1748. Let

(7)

(8)

By the Chain Rule,

(9)

(10)

The wave equation then becomes

(11)

Any solution of this equation is of the form

(12)

where

and

are any functions. They represent two waveforms traveling in opposite directions,

in theNegative

direction and

in the Positive

direction.

The 1-D wave equation can also be solved by applying a Fourier Transform to each side,

(13)

which is given, with the help of the Fourier Transform Derivative identity, by

(14)

where

(15)

This has solution

(16)

Taking the inverse Fourier Transform gives




			(17)

where

(18)

(19)

This solution is still subject to all other initial and boundary conditions.

The 1-D wave equation can be solved by Separation of Variables using a trial solution

(20)

This gives

(21)

(22)

So the solution for

(23)

Rewriting (22) gives

(24)

so the solution for

(25)

where

. Applying the boundary conditions

to (23) gives

(26)

where

is an Integer. Plugging (23), (25) and (26) back in for

in (21) gives, for aparticular value of



			(27)

The initial condition

then gives

, so (27) becomes

(28)

The general solution is a sum over all possible values of

, so

(29)

Using Orthogonality of sines again,

(30)

where

is the Kronecker Delta defined by

(31)

gives


			(32)

so we have

(33)

The computation of

s for specific initial distortions is derived in the Fourier Sine Series section. We already have foundthat

, so the equation of motion for the string (29), with

(34)

(35)

where the

Coefficients are given by (33).

A damped 1-D wave

(36)

given boundary conditions

			(37)
			(38)

initial conditions

			(39)
			(40)

and the additional constraint

(41)

can also be solved as a Fourier Series.

(42)

where

			(43)
			(44)

			(45)

To find the motion of a rectangular membrane with sides of length and (in the absence of gravity ), use the 2-D wave equation

(46)

where

is the vertical displacement of a point on the membrane at position (

) and time

. Use Separation of Variables to look for solutions of the form

(47)

Plugging (47) into (46) gives

(48)

where the partial derivatives have now become complete derivatives. Multiplying (48) by

gives

(49)

The left and right sides must both be equal to a constant, so we can separate the equation by writing the right side as

(50)

This has solution

(51)

Plugging (50) back into (49),

(52)

which we can rewrite as

(53)

since the left and right sides again must both be equal to a constant. We can now separate out the

equation

(54)

where we have defined a new constant

satisfying

(55)

Equations (53) and (54) have solutions

(56)

(57)

We now apply the boundary conditions to (56) and (57). The conditions

and

mean that

(58)

Similarly, the conditions

and

give

and

, so

and

, where

and

are Integers. Solving for the allowed values of

and

then gives

(59)

Plugging (52), (56), (57), (58), and (59) back into (22) gives the solution for particularvalues of

and

(60)

Lumping the constants together by writing

(we can do this since

is a function of

and

, so

can be written as

) and

, we obtain

(61)

Plots of the spatial part for modes (1, 1), (1, 2), (2, 1), and (2, 2) follow.

The general solution is a sum over all possible values of and , so the final solution is

(62)

where

is defined by combining (55) and (59) to yield

(63)

Given the initial conditions and , we can compute the s ands explicitly. To accomplish this, we make use of the orthogonality of the Sine function in the form

(64)

where

is the Kronecker Delta.This can be demonstrated by direct Integration. Let

in (64), then

(65)

Now use the trigonometric identity

(66)

to write

(67)

Note that for an Integer

, the following Integral vanishes


			(68)

since

when

is an Integer. Therefore,

when

. However,

does notvanish when

, since

(69)

We therefore have that

, so we have derived (64). Now we multiply

by two sineterms and integrate between 0 and

and between 0 and

(70)

Now plug in

, set

, and prime the indices to distinguish them from the

and

in (70),


			(71)

Making use of (64) in (71),

(72)

so the sums over

and

collapse to a single term

(73)

Equating (72) and (73) and solving for

then gives

(74)

An analogous derivation gives the

s as

(75)

The equation of motion for a membrane shaped as a Right Isosceles Triangle of length on a side and with the sides oriented along the Positive and axes is given by



	(76)

where

(77)

and

Integers with

. This solution can be obtained by subtracting two wave solutions for asquare membrane with the indices reversed. Since points on the diagonal which are equidistant from the center must have thesame wave equation solution (by symmetry), this procedure gives a wavefunction which will vanish along the diagonal aslong as

and

are both Even or Odd. We must further restrict the modes since those with

givewavefunctions which are just the Negative of

and

give an identically zero wavefunction. The followingplots show (3, 1), (4, 2), (5, 1), and (5,3).

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Wave Equation in Prolate and Oblate Spheroidal Coordinates.'' §21.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 752-753, 1972.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 124-125, 1953.

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