Sound Propagation through Media Nachiketa Tiwari Indian Institute of Technology Kanpur
LECTURE-17 HARMONIC PLANE WAVES
Introduction In this lecture, we discuss propagation of 1-D planar waves as they travel through a lossless, isotropic, homogenous fluid media. In such a media, the velocity of sound, c, is constant throughout the field, and equals (rp 0 /ρ 0 ). Since, the wave in such a case is assumed to be onedimensional and planar, its pressure amplitude and phase is a constant for all points on a plane normal to direction of wave propagation. Ditto for other acoustic variables such as particle velocity, acceleration, etc.
Finally, we reiterate, that for such a wave, whose direction of propagation is aligned to our reference, x, direction, the governing wave equation is: Eq. 17.1
Harmonic Plane Wave In X-Direction Further, if the plane wave, travelling in x-direction, is harmonic in nature, then, the solution for p, can be written as: Eq. 17.2 Here, and as shown later, p+ and P- represent complex amplitudes of the wave travelling in +ve and ve directions, respectively. Further, we can also express the relation for particle velocity u(x, t) for such a wave as: Eq. 17.3 A detailed proof for Eq. 17.3 will be provided later. Also, in Eq. 17.3, Z 0 is known as characteristic impedance and equals, ρ 0 c.
Plane Waves In Arbitrary Direction Next, we explore travel of plane harmonic waves in some arbitrary directions. Since, for such waves, the wave propagation direction and x-axis are not aligned, the governing equation has to same as that for 3-D waves travelling in lossless fluids. Hence, the governing equation for such waves is:
Let us assume that a possible solution for such a plane wave is: Eq. 17.4 This assumed solution satisfies the 3-D wave equation only if the following relation is true: Eq. 17.5
Planar Waves In Arbitrary Direction Now, if there is a planar wave travelling in some arbitrary direction, then its governing equation must provide information on: I. Direction of propagation i.e. the direction in which this wave is travelling. It is on planes normal to such a direction, that the phase and amplitude of acoustic variable is constant. This direction will be later expressed through the propagation vector. II. The position at which the acoustic variable (such as pressure) has to be evaluated. This will be later expressed as position vector. In the case when propagation direction was defined such that it is aligned to x-direction, the direction of position vector and propagation vectors are the same. However for plane waves travelling in arbitrary directions, these directions are different.
Plane Waves In Arbitrary Direction Thus, in next few steps, we will interpret Eqs. 17.5 and 17.4 so as to extract information on propagation and position vectors from them. For location (x,y,z), where wave proportion has to be calculated, the position vector by definition is, such that: Eq. 17.6 where, respectively. are unit vectors in x, y and z directions
Now, let us assume that the propagation vector is defined as: Eq. 17.7 Thus; we note the following: and Eq. 17.8a Eq. 17.8b
Plane Wave In Arbitrary Direction Thus, if Eq. 17.5 holds true, Eq. 17.4 can be written as: Eq. 17.9 In this equation: I. in the wave-number for the wave. II. III. K x / k, K y / k and K z / k are direction cosines of propagation vector. represents the location of point of internet, i.e. position vectors.
Planar Waves In Arbitrary Direction We further note that is the gradient of scalar, and it equals by definition. Further, we know that the direction of this gradient, which equals is normal to surface of constant phase. Hence from all these observations, we can state that: this represents a plane wave travelling in (k x2 +ky 2 +k z2 ) and also represent direction cosines of propagation vector.
Example Given a planar wave moving forward with angular frequency ω, which is travelling in direction (1,2,3) with respect to origin, what is its governing equation? Solution : Direction cosines of propagation vector are 1/ (14), 2/ (14) and 3/ (14). Thus : Likewise, and
But Hence : and Hence the wave propagation Eq. is : where, k x, k y, k z are defined above.
Planer Waves In Arbitrary Direction A special case for planar wave travelling in arbitrary direction would be when its iso-phase planes are parallel to its z-axis. For such a situation, Eq. 17.4 reduces to: Eq. 17.10 Here, surface of constant plane can be expressed as: Eq. 17.11 where ψ is a constant. Equation 17.11 represents planes parallel to z-axis with a slop of -k x /k y as measured in x-y plane.
These planes are shown in Fig. 17.1 :
From figure 17.1 we see that : and Thus, Thus, Eq. 17.12