Three-dimensional sound waves--a classic example of the use of vector calculus--and a word to the wise
Important branches of engineering--electromagnetic fields, fluid mechanics and elasticity--make use of the combination of calculus with vector notation to understand and analyse the behaviour of fields, which are scalar, vector and tensor functions of space and time. This vector calculus originated in the mathematical physics of the 19th century, and remains an essential foundation for the understanding of physics, and its engineering applications. The dynamics of sound waves is an example of this that shows very clearly and straightforwardly the method of approach, in analysing a problem that can be understood in concrete, mechanical terms.
Plane sound waves can be treated by taking the particle displacement as the fundamental quantity. This method involves a minimum of advanced mathematics and a maximum of physical insight. After one is familiar with this procedure, one wishes to treat sound waves of more general shapes, and problems such as radiation, scattering and diffraction. However, the method used for plane waves cannot easily be generalized, because the particle displacement is a vector quantity, and three interrelated components must be considered in the general case. The best course is to start again with a new method of description, one that arises in the study of fluid mechanics, that can more easily be generalized. We will see that we can once again reduce the problem to the determination of a single function of space and time.
Since we will be dealing with the combination of vectors and the calculus of several variables, a few preliminary words on the subject of what is called vector calculus are in order. It is possible to write out explicity the components of our vectors, but we are forced to make a choice of coordinate systems. Usually, rectangular components are used. Whenever any doubt arises, this resolution into components can be used to remove it. However, we are used to using symbols that stand for all the vector components, in whatever system of coordinates they may be expressed, such as a vector r, distinguished by bold face in type, or by drawing an arrow over it in handwritten text. We know how to form the scalar (dot) and vector (cross) products of two vectors, and how to interpret them intuitively, without referring to a particular set of components. Vectors reduce bookkeeping to a minimum. Vectors can be added or subtracted, or multiplied by a scalar, and we can think of a vector as the product of a magnitude r and a unit vector n, r = rn. A unit vector is, of course, one whose magnitude is unity, and is thought of as carrying pure direction. When we need coordinates explicitly, we can use the unit vectors i,j,k in the directions of the axes of x,y,z. All this is carried over into vector calculus, making a compact mode of expression that makes it easy to understand complicated meanings and relations in an abstract sense.
Differentiation is represented by the operator nabla or del, an inverted Δ. Since del is not easy to display here, I shall use the word, but in all illustrations it will receive its usual symbol. Del is the operator shown in the Figure below (where all the equations referred to in the text can be found). When it operates on a scalar function f(x,y,z), it is called the gradient, or grad f. A single function, like f, is called a scalar because it has no vector character, but can change the length (scale) of any vector it multiplies. The gradient of f is a vector pointing in the direction in space in which f increases the most rapidly, and its scalar product with a unit vector n, n.grad f, is the rate of change with distance in that direction. grad f represents the three gradients of the three component functions of the vector field f, and is a tensor, not a vector. The scalar product of a tensor and a vector is another vector, so a tensor relates one vector to another. Pressure is a very familiar tensor, and an extremely simple one, simply the pressure p times a unit tensor I, pI. I has the property that the scalar product of any vector into I is just that vector again. The general stress tensor relates a vector force per unit area to a direction, the normal to the area. pI is just a very simple stress tensor, an isotropic one.
Del can operate on a vector in two ways, corresponding to the two kinds of products of vectors, dot and cross. The scalar product with a vector function f gives a scalar, called div (divergence) f, and the vector product gives another vector, called curl (or rot) f. These functions have important physical interpretations that depend on what is being described. The curl of the velocity is called the circulation, for example. If f is the velocity of the particles of a flowing fluid, then the divergence is the rate at which fluid flows out of a small volume per unit volume, and the curl is the line integral of the velocity around a small curve per unit area enclosed by the curve (roughly speaking). An important property is that the curl of the gradient of a function always vanishes. Inversely, if the curl of a vector vanishes in a region, the vector can be expressed as the gradient of a scalar. The divergence of the curl of a vector function also vanishes identically. The divergence of a gradient of a function is the sum of the second derivatives, called the Laplacian, and expresses a kind of curvature in three dimensions. The divergence of the curl always vanishes.
The divergence and curl express microscopic, local properties of a vector field (dilatation and rotation), that correspond to the macroscopic properties of the surface integral of the field over a closed surface (called the flux), or the line integral of the vector around a closed curve. The relation between divergence and flux is called the divergence theorem, shown in the Figure, where the vector field involved is the velocity field of a fluid. On the right-hand side is the surface integral over a closed surface of the normal component of the velocity. This is the volume of fluid leaving the volume per unit time. On the left-hand side, we have the volume integral of a quantity called the divergence of v, which expresses the rate of creation of fluid throughout the interior of the surface. Of course, these two quantities must be equal. The divergence is just the flow out of a small element with sides dx,dy,dz. It is practically the definition of the time derivative of the relative change in volume Δ, the dilatation. The relation between the curl and the line integral is called Stokes' Theorem, and says that the line integral about a closed curve is equal to the surface integral of the curl over any surface bounded by the curve. We will not need it here, so it is not explicitly shown. We have now presented the heart of vector calculus, and refer the reader to texts for more examples and more of the theory. Vector calculus has been a notorious stumbling block for student engineers.
The motion of any fluid can be specified by giving the particle velocity as a function of space and time, that is, by three functions v(x,y,z,t), giving a vector, the velocity, at any point (x,y,z) as a function of the time t. This is a vector field, as shown in the Figure. The particle of fluid now at (x,y,z) will be at (x+udt,y+vdt,z+wdt) a short time dt later, and so on, where u,v,w are the components of the velocity. In this way, the path of each particle is strictly determined. The change in velocity of the particle of fluid is composed of two parts, the change in velocity as a function of time, and the change in velocity because the particle is at a new position at a later time. The same is true for the change of any property of the particle, so we define a substantial derivative, and denote it by straight d's rather than by the curved d's of partial differentiation. This concept is of great importance in fluid mechanics.
The force on a small element of fluid of sides dx,dy,dz is -del p, where p is also a field, p(x,y,z,t), as shown in the figure on the left for the x-component. We can now use Newton's second law to write the equation of motion for the velocity, as shown. This is a quite general equation, applying to general fluid flows, so long as pressure is the only force acting. For sound waves of small amplitude, we make the assumptions shown, which amount to a neglect of the products of small quantities, and lead to linear equations. The third equation, the linearized equation of state, merely says that the change in pressure is proportional to the relative change in density s (this is the condensation (ρ - ρ0)/ρ0). The constant of proportionality κ is called the bulk modulus, and has the dimensions of pressure, since condensation is a dimensionless ratio (which is a very good thing). In an ideal gas it is indeed simply the pressure, for an isothermal process, or γ (ratio of specific heats) times the pressure for an adiabatic process. Now the equation of motion takes on a much simpler form. We must express s in terms of v and eliminate one of the quantities so that the equation can be solved. The result is just that the time derivative of the condensation is the negative divergence of the velocity. We have already met the divergence of the velocity, which is strictly the time derivative of the relative increase in volume, but for small quantities this is the negative condensation.
If s is eliminated, we find the velocity equation, which is not quite the wave equation that we are expecting. (Note that del does not commute with the Laplacian!) Therefore, the particle velocity does not, in general, satisfy the simple wave equation. It is no wonder that it, or the related displacement, are not the best ways to describe the motion. If v is eliminated instead, we do find the scalar wave equation for the condensation. The condensation will, therefore, be a suitable quantity to describe general waves. It has the disadvantage, however, that it is not simple to find the velocity when the condensation is known. In general, one must solve an additional differential equation to do so, or use a subterfuge.
One way around this is to note that the linearized wave equation shows that the time derivative of the rotation, curl v (del cross v) vanishes. Now, before the sound wave arrives, curl v is certainly zero, since v is zero everywhere. Since it does not change with time, it must remain zero. In this case, v can be expressed as the negative gradient of a scalar quantity, called the velocity potential φ, a result well known in fluid mechanics. The negative sign is inessential, but is conventionally included by analogy with electrostatics. If this result is substituted in the velocity equation, we find that the velocity potential satisfies the wave equation, as desired, provided we integrate the equation from some arbitrary starting point, and set the arbitrary constant of integration equal to zero. This is the same as naively cancelling the dels on the two sides. Now we have a scalar quantity that obeys the wave equation from which we can easily find the particle velocity. Moreover, by applying the same treatment to the linearized equation of motion, the condensation can be expressed as the time derivative of the vector potential, so all the important quantities of the wave can be found by differentiating the velocity potential.
The fact that the rate of change of the circulation is zero is a consequence of the linearized wave equation, not of the exact equations of motion. This must be kept in mind when waves of finite amplitude and other circumstances arise. In fact, we will meet such a case later in this paper.
The various expressions containing del can be expressed in other coordinate systems, such as cylindrical and spherical, that are appropriate for these symmetries. This comes perilously close to bookkeeping, so I shall not do it here, merely remark that reference works give expressions for the various operators in these other coordinate systems. The case of isotropic spherical waves is a specially simple case that is instructive to carry out from first principles, however. We can use the definition of dilatation to get an expression for the equation of continuity, then combine it with the equation of motion to find a wave equation for spherical waves. This wave equation is just like the one-dimensional wave equation that holds for plane waves, except that the space variable is now r, and the quantity that satisifies the simple wave equation is the product of r and the velocity potential. The equations on the right show the reasoning. The equation of continuity, you will notice, is essentially the divergence theorem, where we express the flux out of a thin spherical shell in two different ways. The wave equation immediately gives the typical 1/r dependence of the wave amplitude that is necessary for conservation of energy. The general solution is rφ= f(ct - r) + F(ct + r), where f and F are two arbitrary functions, the first representing an outgoing wave, the second an incoming wave. Taking an outgoing wave only, the radial velocity is v = f'/r + f/r2. The first term predominates for large distances, the second for small distances, from the origin. The amount of fluid entering at the origin is 4πf(ct), which represents a monopole source (such as a siren or an explosion). The condensation, on the other hand, is just f'/c2r, so that the product rs propagates without change (of course! - it satisfies the simple wave equation).
We can construct new solutions to the wave equation by superimposing old ones. Suppose two monopole sources of equal but opposite strengths f and -f are a distance b apart. Let r be the distance from the midpoint between the two sources to any point, and let θ be the angle this radius vector makes with the axis of the sources. Then, if s is much smaller than r, we find φ = fb cos θ / r2. This is the field of a dipole source, more typical of the usual sources of sound, in which a net amount of fluid is not introduced or withdrawn, but things are just moved about. We could carry on in this way to assemble higher-order sources, but the dipole source shows the general idea.
A curious phenomenon may be explained here. If there is an underwater explosion (for example) close to the surface, the effect of the surface must be taken into account (it reflects the sound). The boundary condition is that the condensation, and so the time derivative of the velocity potential must vanish on the surface (the air cannot resist). The superposition of the original source and a so-called image source of opposite phase located where a mirror image in the surface would be will satisfy the boundary conditions for the wave in the water, since the surface is equal distances from the two sources. From any great distance, the source and its image look like a double source. A double source does not radiate normal to its axis, however, (cos θ = 0) so at small depths, the explosion will not be heard. The direct wave is cancelled by the wave reflected from the surface.
It is best to keep in mind that all solutions obtained by superposition are strictly true only for waves that obey the simple wave equation, and all solutions based on boundary conditions at an interface are true only as long as the interface is ideal.
Using the velocity potential, we can investigate the reflected and refracted plane waves when a plane wave is incident on an ideal plane interface between two media. The method is to assume a superposition of incident and reflected waves in the first medium, and a transmitted wave in the second medium, and then determine the amplitudes of the waves that satisfy the boundary conditions at the interface. We did this in another place for plane waves normally incident on an interface using the particle displacement to describe the waves. Now we can consider a wave incident at any angle. Since the wavefronts must move in step on the interface, the waves are chosen to obey the usual laws of reflection and refraction. In the diagram at the right, the waves are described by their wave vectors k. n is the normal to the interface. Therefore, k.n = k cos θ, k'.n = -k cos θ, and k".n = k" cos θ" are the components of k in the direction of the normal.
At the interface, the pressure and the normal component of the particle velocity must be continuous. If the pressure is not continuous, there will be a finite force on an infinitesimal mass, which is impossible, and if the normal velocity is not continuous, the two media will pull apart or penetrate, which likewise cannot happen. These conditions give the equations shown in the Figure, where common factors have been eliminated. We have two equations in the two unknown amplitudes, which are easily solved to give the results shown. Note that we have expressed the solutions in terms of the product of the density and the velocity potential, which is proportional to the pressure. Therefore, the squares of the coefficients, corrected for the ratio of beam widths (cos θ"/cos θ for the transmitted wave, unity for the reflected), are the intensity reflection and refraction coefficients. These equations can be found in the literature.
Hang on a minute. There is also a velocity component parallel to the interface, and if it is not continuous, the two media will slide on one another. It's easy to see the condition on the velocity potential is φ + φ' = φ" (or differing by a constant), which is not consistent with the other boundary conditions, unless there is no density difference between the media. Therefore, in the solution we have just obtained this component of the velocity is not continuous across the boundary, in general. In neglecting it, we are assuming that the fluid cannot support a viscous shearing stress. However, a fluid can indeed do so; consider the interface between oil and water, for example. Our solution is, therefore, incomplete, and what happens at the interface is more interesting. Significantly, Lamb does not consider this problem. In the analogous problem in electromagnetism, all the boundary conditions are satisfied by a proper choice of reflected and refracted waves.
This third boundary condition is not as 'essential' as the other two, and may not have a large effect on the problem. Indeed, the velocities are small in any case. However, circulation is created in case of a mismatch, and this is incompatible with the existence of a rigorous velocity potential. Some dissipation of energy must result, though the angular relations of the waves will not be affected. An additional disturbance must be created at the interface which our analysis is not capable of explaining. In the case of an interface between solid media, transverse waves also enter, and yield a satisfactory solution by matching boundary conditions, without surface dissipation. It might be interesting to look for effects of this kind. For our purposes, it shows the danger of proceeding with a formal analysis without understanding, as students so love to do.
Composed by J. B. Calvert
Created 14 May 2000
Last revised 21 May 2000