Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions. Among them are the well-known d'Alembert, Poisson, and Kirchhoff representation formulae in low space dimensions.
1. Introduction
The wave equation for a function of space variables and the time is given by
where
is the Laplacian. The wave equation is encountered often in applications. For the equation can represent sound waves in pipes or vibrations of strings, for waves on the surface of water, for waves in acoustics or optics. Therefore, formulae that give the solution of the Cauchy problem in explicit form are of great significance. In the Cauchy problem (initial value problem) one asks for a solution of (1.1) defined for , that satisfies (1.1) for , and the initial conditions
If and , , then the classical solution of problem (1.1), (1.3) is given by d’Alembert’s formula
If and , , then the solution of problem (1.1), (1.3) is given by Poisson’s formula
where , , and .
If and , , then the solution of problem (1.1), (1.3) is given by Kirchhoff’s formula
where , , , and is the surface element of the sphere .
Passing to an arbitrary let us denote by the solution of the problem
It is easy to see that then the function
is the solution of the problem
Indeed, integrating (1.7) we get
Hence,
by the second condition in (1.8). On the other hand, from (1.9),
Comparing (1.13) and (1.14), we get (1.10). Besides,
so that initial conditions in (1.11) are also satisfied.
Consequently, the solution of problem (1.1), (1.3) is represented in the form
It follows that it is sufficient to know an explicit form of the solution of problem (1.7), (1.8). It is known [1, 2] that
where , , , and is the surface element of the sphere .
In the present paper, we give a new proof of formulae (1.17), (1.18) for the solution of problem (1.7), (1.8). Our method of the proof is based on the spectral theory of the Laplace operator. We hope that such a method may be useful also in some other cases of the equation and space.
The paper consists, besides this introductory section, of three sections. In Section 2, we describe the structure of arbitrary rapidly decreasing function of the Laplace operator, showing that it is an integral operator and giving an explicit formula for its kernel. Next we use these results in Section 3 to derive the explicit representation formulae for the classical solution to the initial value problem for the wave equation in arbitrary dimensions. The final Section is an appendix and contains some explanation of several points in the paper.
2. Structure of Arbitrary Function of the Laplace Operator
Let be the self-adjoint positive operator obtained as the closure of the symmetric operator determined in the Hilbert space by the differential expression
on the domain of definition that is the set of all infinitely differentiable functions on with compact support. Let denote the resolution of the identity (the spectral projection) for A:
Next, let be any infinitely differentiable even function on the axis with compact support and
its Fourier transform. Note that the function tends to zero as faster than any negative power of . Consider the operator defined according to the general theory of self-adjoint operators (see [3]):
The following theorem describes the structure of the operator showing that it is an integral operator and giving an explicit formula for its kernel in terms of the function .
Theorem 2.1. The operator is an integral operator
Further, there is a smooth function defined on the interval such that
The function depends on the function as follows. If one sets
then
where denotes the th order derivative of . Further, if , then . For any solution of the equation
the equality
holds for .
Proof. First we consider the case . In this case, the statements of the theorem take the following form: for ; the operator is an integral operator of the form
and for any solution of the equation
the equality
holds. To prove the last statements note that, in the case , the operator is generated in the Hilbert space by the operation and the operator by the operation . The resolvent of the operator has the form
while the spectral projection of the operator has the form (see [3, page 201])
Therefore,
where we have used the inversion formula for the Fourier cosine transform. Therefore, (2.11) is proved. To prove (2.13) note that the general solution of (2.12) is
where and are arbitrary constants. Then, we have, for ,
where we have used the fact that the function is even and therefore
The same result can be obtained similarly for . Thus, (2.13) is also proved. Now we consider the case . We shall use the integral representation
of the resolvent of the operator . As is known [4, Section 13.7, Formula (13.7.2)],
where is the Hankel function of the first kind of order . Next, according to the general spectral theory of self-adjoint operators [3, page 150, Formula (11)], we have
Therefore, from (2.4) it follows that the representation (2.5) holds with
Now the representation (2.6), which expresses that is a function of , follows from (2.23) by (2.21). To prove (2.10) we use (2.23). By virtue of (2.23),
see Appendix. Here we have used the fact that from (2.9) it follows that
that is,
and therefore
Finally, to deduce the explicit formulae (2.7), (2.8), we take in (2.10). Then, putting , we can write
If we set
then the left-hand side of (2.28) equals
On the other hand,
where
is the surface area of the -dimensional unit sphere ( is the gamma function) and denotes the surface element of the sphere . Therefore, setting
we get that (2.28) takes the form
Substituting here the expression of given in (2.29) and making then the change of variables , we obtain
Hence (2.7) follows. Further, it is not difficult to check that the formula (2.33) for is equivalent to (2.8), see Appendix. Since is smooth and has a compact support, it follows from (2.7), (2.8) that the function also is smooth and has a compact support; more precisely, if , then . This implies, in particular, convergence of the integral in (2.10) for each fixed . The theorem is proved.
Consider the Cauchy problem (1.7), (1.8):
where , , , .
For , , let us set
Since
applying (2.9), (2.10), we get
Hence, by the inverse Fourier transform formula,
Multiplying both sides of the last equality by and then integrating on , we get
Substituting here for its expression
and setting
we obtain
Obviously, the function defined by (3.9) is the solution of problem (3.1), (3.2). Next we will transform the left-hand side of (3.10) using Theorem 2.1.
First we consider the case . In this case, (3.10) takes the form
and from (2.7), (2.8) we have
Therefore, making the change of variables and taking into account the evenness of the function , we can write
Substituting this in the left-hand side of (3.11), we obtain
Hence, by the arbitrariness of the smooth even function with compact support, we get
Further assume that . Making the change of variables
where is the surface element of the unit sphere , we get
Further, making in the right-hand side of (3.17) the change of variables
where is the surface element of the sphere , we have
Therefore,
and (3.10) becomes
Consider the cases of odd and even separately.
Let . Then, by (2.8) we have
and it follows from (2.7) (by successive differentiation) that
Therefore,
and (3.21) takes the form
Further, integrating times by parts, we get
where
Since is identically zero for large values of , we have from (3.27) that . Also, it follows directly from (3.27) that . Therefore, (3.25) becomes
Since in (3.28) is arbitrary smooth even function with compact support, we obtain that
This coincides with (1.17) by (3.19).
Now let us consider the case . In this case, by (2.8) we have
and therefore
Hence, setting
we get
Substituting this in the left-hand side of (3.21) (beforehand replacing by in the left side of (3.21)), we obtain
or, using (3.23),
Further, integrating times by parts, we get
where
Since is identically zero for large values of , we have from (3.37) that . Also, using the expression of ,
we can check directly from (3.37) that . Therefore, (3.35) becomes
Since in (3.39) is arbitrary smooth even function with compact support, we obtain that
This coincides with (1.18) by (3.32).
Appendix
For reader’s convenience, in this section we give some explanation of several points in the paper.(1)Let us show how (2.33) for implies (2.8).
Let , where . Then, since
Equation (2.33) takes the form
Hence applying the differentiation formula
repeatedly, we find
which gives (2.8) for .
In the case with , (2.33) takes the form
Hence,
Therefore, taking into account that by virtue of
we have
we get
In the right-hand side we replace by , then divide both sides by and integrate on to get
because for any , using the change of variables , we have
Therefore, differentiating (A.10) with respect to , we get
Thus, (2.8) is obtained also for with .
(2) Here we explain (2.24). Note that since the spectrum of the operator is (zero is included into the spectrum), the spectral representation formula (2.4) should be understood in the sense of the formula
where is an arbitrary positive real number and the integral does not depend on ( is zero on because is a positive operator). Therefore, for (2.24) we have to show that
for any .
Since for any
we have
Given , we can choose a such that
since the function is continuous for (we choose the continuous branch of the square root for which ). Further, we choose a number such that
for sufficiently large positive number . This is possible by (2.3) and the fact that has a compact support. Let us set . Then,
Further,
For fixed , the last expression tends to zero as ; hence, and by (A.16), (A.19), and (A.20) we get (A.14).
(3) The formula (2.14) follows from (2.21) for noting that
(4) The difference between operators (formulae (1.17), (1.18)) and (formula (3.25)) is given by
(5) The explicit formula for the solution of the wave equation in the case even can be derived from the case odd by a known computation called the “method of descent” (see [1]).
(6) Since for , , we have , and on the left-hand side of (2.10) the integral is taken in fact over the ball , for fixed . Therefore, this integral is finite for each and any solution of (2.9). We proved (2.10) for . If the solution is an analytic function of , then (2.10) will be held also for complex values of by the uniqueness of analytic continuation.
References
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V. I. Smirnov, A Course of Higher Mathematics, Vol. II, Addison-Wesley, Reading, Mass, USA, 1964.
M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Reidel, Dordrecht, The Netherlands, 1987.
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