3.3 Four Basic Traveling Waves |
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3.3Four Basic Traveling Waves
Because sound pressure p(x, t) is a real physical quantity, the complex solution p(x, t) must be real after applying boundary and initial conditions that are also real numbers. Therefore, the real solution of sound pressure p(x, t) must be constructed by two complex conjugate pairs as a result of the following constraints:
A2B2 ¼ ðA1B1Þ
A1B2 ¼ ðA2B1Þ
or:
A1B1 ¼ A2B2 A
A1B2 ¼ A2B1 Aþ
α1 þ β1 ¼ α2 β2 θ α2 þ β1 ¼ α1 β2 θþ
where we define four new real variables A , A+ ,θ , and θ+ with a subscript “-” to indicate backward waves and a subscript “+” to indicate forward waves.
Based on the above four constraints and four new real variables, the general solution p(x, t) is a real number because of the addition of the following two complex conjugate pairs:
p(x, t)¼ |
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A e jðωtþkxþθ Þ þ A e jðωtþkxþθ Þ |
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e jðωt kxþθþÞ |
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e jðωt kxþθþÞ |
(Form 3 : CEP) |
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þh |
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Note that there are four real unknown variables A , A+ ,θ , and θ+that can be determined by two boundary conditions and two initial conditions. The first two terms above represent a backward wave p (x, t), and the last two terms represent a forward wave p+(x, t).
The general wave solution p(x, t) is the addition of a backward wave p (x, t) and a forward wave p+(x, t):
p(x, t) ¼ p (x, t) + p+(x, t)
¼ A cos (ωt + kx + ϕ ) + A+ cos (ωt kx + ϕ+) (Form 2 : REP)
where the backward wave p (x, t) and the forward wave p+(x, t) can be expressed in Form 1 and Form 2 as:
p (x, t) ¼ A cos (ωt kx + ϕ ) |
(Form 2 |
: REP) |
¼ A c cos (ωt kx) A s sin (ωt kx) |
(Form 1 |
: RIP) |
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3 Solutions of Acoustic Wave Equation |
Note that when ωk ¼ c, it can be shown that every term in the general solutions in both Form 1 and Form 2 satisfies the one-dimensional acoustic wave equation below:
∂2 pðx, tÞ ¼ 1 ∂2 pðx, tÞ ∂x2 c2 ∂t2
In (Form 1:RIP), the sound pressure is formulated with four unknown variables A c, A+c, A s, and A+s as:
p(x, t) ¼ p+(x, t) + p (x, t) |
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¼ A+c cos (ωt kx) A+s sin (ωt kx) |
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þA c cos ðωt þ kxÞ A s sin ðωt þ kxÞ |
(Form 1 : RIP) |
The four basic traveling waves (BTW) are defined as: |
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Four Basic Traveling Waves (BTW) in Form 1
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Cosine Function |
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Sine Function |
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Forward Wave |
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cos |
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sin |
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Backward Wave |
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cos |
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sin |
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The sound pressure in four equivalent forms is shown below for easier reference:
p(x, t)
p(x, t) p(x, t)
p(x, t)
¼A+c cos (ωt kx) A+s sin (ωt kx) +A c cos (ωt + kx) A s sin (ωt + kx)
¼A+ cos (ωt kx + ϕ+) + A cos (ωt + kx + ϕ )
¼12 Aþe jðωt kxþϕþÞ þ 12 Aþe jðωt kxþϕþÞ
þ12 A e jðωtþkxþϕ Þ þ 12 A e jðωtþkxþϕ Þ
¼12 ðAþc þ jAþsÞe jðωt kxÞ þ ðAþc jAþsÞe jðωt kxÞ
þ12 ðA c þ jA sÞe jðωtþkxÞ þ ðA c jA sÞe jðωtþkxÞ
(Form 1:RIP) (Form 2:REP)
(Form 3:CEP)
(Form 4:CIP)
As shown above, the BTWs can also be expressed in four equivalent forms introduced in Chap. 1.
3.4Four Basic Standing Waves
A standing wave is the addition of two traveling waves with the same amplitude, traveling in the opposite direction. Therefore, it is required that:
3.4 Four Basic Standing Waves |
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Aþ ¼ A ¼ A |
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With the above standing wave condition, a standing wave can be depicted as: |
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psðx, tÞ ¼ pþðx, tÞ þ p ðx, tÞ
¼ A cos ωt kx þ ϕþ þ A cos ðωt þ kx þ ϕ Þ
where A, ϕ+, and ϕ are real unknown constants that can be determined by given boundary and initial conditions.
The above equation can also be expressed as: |
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psðx, tÞ ¼ 2A cos hωt þ |
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i cos hkx þ |
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ϕ þϕþ |
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ϕ ϕþ |
(Form 2 : REP) |
The standing wave equation above can be simplified using two new variables, ϕt
and ϕx, to become: |
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ps(x, t) ¼ 2A cos [ωt + ϕt] cos [kx + ϕx] |
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(Form 2 : REP) |
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where the two new variables, ϕt and ϕx, are defined as: |
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ϕ þ ϕþ |
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ϕ ϕþ |
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Expand the general standing wave ps(x, t) above to get (Form 1 : RIP) from (Form 2 : REP) as follows:
ps(x, t) ¼2A cos [ωt + ϕt] cos [kx + ϕx] |
(Form 2 : REP) |
¼2A[cos(ωt) cos (ϕt) |
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sin (ωt) sin (ϕt)][cos(kx) cos (ϕx) |
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sin (kx) sin (ϕx)] |
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¼2A cos (ϕt) cos (ϕx) cos (ωt) cos (kx) + 2A sin (ϕt) sin (ϕx) sin (ωt) sin (kx)
2A sin (ϕt) cos (ϕx) sin (ωt) cos (kx)2A cos (ϕt) sin (ϕx) cos (ωt) sin (kx)
¼Acc cos (ωt) cos (kx) + Ass sin (ωt) |
(Form1 : RIP) |
sin (kx) + Asc sin (ωt) cos (kx) + Acs cos (ωt) sin (kx) |
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where: |
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Acc ¼ 2A cos ðϕtÞ cos ðϕxÞ; |
Ass ¼ 2A sin ðϕtÞ sin ðϕxÞ |
Asc ¼ 2A sin ðϕtÞ cos ðϕxÞ; |
Acs ¼ 2A cos ðϕtÞ sin ðϕxÞ |
where each term of the above equation is a basic standing wave (BSW):
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3 Solutions of Acoustic Wave Equation |
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Four basic standing waves (BSWs): |
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psðx, tÞ ¼ cos ðωtÞ cos ðkxÞ |
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psðx, tÞ ¼ sin ðωtÞ sin ðkxÞ |
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psðx, tÞ ¼ sin ðωtÞ cos ðkxÞ |
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psðx, tÞ ¼ cos ðωtÞ sin ðkxÞ |
BSWs can be expressed by the multiplication of a spatial function and a temporal function. In addition, each BSW can be expressed by the addition of a forward traveling and a backward traveling with the same amplitude as:
psðx, tÞ ¼ cos ðωtÞ cos ðkxÞ ¼ |
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ð cos ðωt þ kxÞ þ cos ðωt kxÞÞ |
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psðx, tÞ ¼ sin ðωtÞ sin ðkxÞ ¼ |
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ð cos ðωt kxÞ cos ðωt þ kxÞÞ |
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psðx, tÞ ¼ sin ðωtÞ cos ðkxÞ ¼ |
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ð sin ðωt þ kxÞ þ sin ðωt kxÞÞ |
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psðx, tÞ ¼ cos ðωtÞ sin ðkxÞ ¼ |
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ð sin ðωt þ kxÞ sin ðωt kxÞÞ |
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The proofs of the above four equations are in the problems.
When the arguments of sine and cosine functions include phases ϕt, ϕx, ϕ+, and ϕ , since cosine and sine only differ by a phase of π/2, all sine functions can be changed to cosine functions, and the use of only cosine functions is enough to represent the standing waves.
Example 3.3 Prove that the following relationship between a standing wave and the combination of a forward wave and a backward wave is true:
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þ cos ðωt þ kx þ |
ϕ Þ |
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cos |
ωt kx þ ϕþ |
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¼ |
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2 cos |
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ωt |
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cos kx |
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Example 3.3 Solution Compare the given equation:
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þ cos ðωt þ kx þ |
ϕ Þ |
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cos |
ωt kx þ ϕþ |
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¼ |
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þ |
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2 cos |
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ωt |
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þ þ |
cos kx |
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to the following trigonometric property: