3.5 Conversion Between Traveling and Standing Waves

64 3 Solutions of Acoustic Wave Equation

Traveling waves p (x, t) can be categorized into four basic traveling waves (BTW) expressed in (Form 2 : RIP) as:

pþðx, tÞ ¼ Aþc cos ðωt kxÞ pþðx, tÞ ¼ Aþs sin ðωt kxÞ p ðx, tÞ ¼ A c cos ðωt þ kxÞ p ðx, tÞ ¼ A s sin ðωt þ kxÞ

or:

Four Basic Traveling Waves (BTW) in Form 1

When ωt and kx have different signs, such as (ωt kx), it is a forward traveling wave. When ωt and kx have the same signs, such as (ωt + kx), it is a backward traveling wave.

Because the above four BTWs are independent functions, any traveling wave can be constructed from these four BTWs. Because these four BTWs are simple, yet can fully represent any traveling wave, these four BTWs will be used in acoustic analysis throughout this course.

A standing wave can be constructed with two traveling waves that have the same amplitude and are traveling in the opposite direction.

For example, choose a forward traveling wave p+(x, t) and a backward traveling wave p (x, t) as follows:

pþðx, tÞ ¼ Aþc cos ðωt kxÞ

p ðx, tÞ ¼ A c cos ðωt þ kxÞ

Assume these two traveling waves have the same amplitude as:

Aþc ¼ A c A

It can be shown that the addition of these two traveling waves is a standing wave ps(x, t) as:

psðx, tÞ ¼ pþðx, tÞ þ p ðx, tÞ

¼A cos ðωt kxÞ þ A cos ðωt þ kxÞ

¼A ½ cos ðωtÞ cos ðkxÞ þ sin ðωtÞ sin ðkxÞ& þA ½ cos ðωtÞ cos ðkxÞ sin ðωtÞ sin ðkxÞ&

¼2A cos ðωtÞ cos ðkxÞ

Note that the two traveling waves must have the same amplitude and travel in opposite directions to construct a standing wave.

Example 3.4 The following is a harmonic function p(x, t) of one-dimensional sound pressure in Cartesian coordinates:

hi

pðx, tÞ ¼ Re P cos ðkxÞe jðωtÞ

where P is a real number.

(a)Show that the harmonic function p(x, t) can be constructed by multiplying two trigonometric cosine functions:

pðx, tÞ ¼ P cos ðkxÞ cos ðωtÞ

(b)Show that the harmonic function p(x, t) can be constructed by adding two trigonometric cosine functions:

pðx, tÞ ¼ P2 ½ cos ðωt þ kxÞ þ cos ðωt kxÞ&

Note that each term in the above function has no phase and can be considered as either Form 1 or Form 2.

Example 3.4 Solution

(a) The given harmonic function p(x, t) is:

hi

pðx, tÞ ¼ Re P cos ðkxÞe jðωtÞ

Since both P and cos(kx) are real numbers, we can move these real parts to outside of the “Re” function as:

h i pðx, tÞ ¼ P cos ðkxÞ ∙ Re e jðωtÞ

For comparing the given forward and backward waves to the above formula, convert the sine function of the given forward wave p+(x, t) to a cosine function as:

Comparing the converted forward and backward waves to the formulas and letting:

b)The locations of the peaks and valleys in terms of the wavelength are:

Peak at kx ¼ 34π ! 2λπ x ¼ 34π ! x ¼ 38λ

Valley at kx ¼ 74π ! 2λπ x ¼ 74π ! x ¼ 78λ