Wave Mechanics: Mechanical Wave Impedance

WM18–WM20 developed energy and power transport on an ideal string, and WM21 generalized the idea of energy flux to higher-dimensional waves. We now return to the one-dimensional string and ask a different but closely related question:

How much transverse force is associated with a given transverse velocity in a traveling wave?

The answer is the characteristic mechanical wave impedance of the string.

For an ideal string with Tension T, linear mass density μ, and wave speed

(1)

the characteristic impedance is

(2)

Its SI unit is

(3)

The impedance Z₀ is a property of the medium and its tension. It tells us how transverse force and transverse velocity are related for a one-way traveling wave [1, 2, 3, 6].

1 Mechanical impedance and characteristic impedance

In vibration theory, a general mechanical impedance is often defined in harmonic steady state as a ratio of force amplitude to velocity amplitude,

(4)

For a mass-spring-damper system or another reactive load, this quantity can depend on frequency and can be complex.

A lossless uniform string has a particularly simple traveling-wave result. Its characteristic impedance

(5)

is real and, within the ideal nondispersive string model, independent of frequency.

This article uses Z₀ for that traveling-wave property. The ratio of force to velocity in an arbitrary standing-wave field need not equal Z₀ point by point.

2 Force and velocity at a cut in the string

Let u(x,t) be the transverse displacement. The local transverse velocity is

(6)

For small slopes, define the transverse force transmitted across a cut in the positive x direction as

(7)

With this sign convention, the instantaneous power flowing in the positive x direction is exactly the WM19 result

(8)

The sign convention is useful because positive P means energy transport toward increasing x.

3 Right-moving wave: derive the impedance

For a right-moving profile

(9)

WM15 showed that

(10)

Therefore

(11)

The transmitted transverse force becomes

F⊥(+x)	= −Tu x	(12)
	= ut.	(13)

Define

(14)

Then a right-moving wave satisfies

(15)

Because

(16)

we may rewrite the impedance in two equivalent forms:

Z0	=	(17)
	= ,	(18)

and

(19)

Thus

(20)

4 Left-moving wave and the sign of power

For a left-moving profile

(21)

we have

(22)

Hence

(23)

and

F⊥(+x)	= −Tu x	(24)
	= −ut	(25)
	= −Z0ut.	(26)

Therefore

(27)

The medium has the same positive characteristic impedance Z₀ in either direction. The sign change records the direction of power flow.

5 Power written in impedance form

For a right-moving wave,

P	= F⊥(+x)u t	(28)
	= Z0ut2.	(29)

Thus

(30)

For a left-moving wave,

(31)

This reproduces the directional power result from WM19 because

(32)

The impedance language therefore compresses the force, velocity, and power relations into a compact set of equations.

6 Sinusoidal average power

Consider a right-moving sinusoidal wave

(33)

Its transverse velocity is

(34)

The RMS transverse velocity is

(35)

Averaging

(36)

over one cycle gives

(37)

Equivalently,

(38)

Since Z₀ = μc, this is exactly the WM20 formula

(39)

This is the mechanical-wave analogue of the familiar statement that power depends on the square of a wave amplitude multiplied by a characteristic impedance or admittance factor.

7 How T and μ affect speed and impedance

The speed and impedance depend differently on tension and linear density:

(40)

Thus increasing tension with μ fixed increases both c and Z₀ as .

Increasing linear density with T fixed has opposite effects on the two quantities:

(41)

This distinction becomes central at an interface.

8 Why impedance matters at an interface

Suppose a harmonic wave traveling in medium 1 reaches an ideal junction with medium 2. Let the characteristic impedances be

(42)

At an ideal massless junction, two conditions are imposed:

1. the transverse displacement is continuous;
2. the transverse force is continuous.

Write the incident, reflected, and transmitted displacement amplitudes as

(43)

Displacement continuity gives

(44)

For harmonic waves, the transverse-force condition gives

(45)

The minus sign occurs because the reflected wave travels in the negative x direction.

Solving the two equations gives the displacement-amplitude reflection coefficient

(46)

and the displacement-amplitude transmission coefficient

(47)

These formulas use the displacement-amplitude convention. Other wave variables can have different amplitude-coefficient formulas even though the physical power balance is the same [1, 2, 3].

9 Power reflection and transmission

For a harmonic traveling wave, the magnitude of the average power is

(48)

The fraction of incident power reflected is therefore

(49)

Using the transmitted amplitude coefficient t,

(50)

Substituting the expression for t gives

(51)

For a lossless ideal junction,

(52)

Note that t itself can exceed 1 without violating energy conservation. Power depends on both amplitude and impedance.

10 Matched, fixed-like, and free-like limits

The reflection coefficient can be written in terms of the impedance ratio

(53)

as

(54)

Three limits are especially important.

Matched impedance

If

(55)

then

(56)

No reflected wave is required.

Very large terminating impedance

If

(57)

then

(58)

The displacement reflection is inverted, reproducing the fixed-end behavior introduced in WM12.

Very small terminating impedance

If

(59)

then

(60)

The displacement reflection is not inverted, reproducing the free-end behavior.

Thus the fixed and free boundaries from WM12 can be understood as limiting cases of an impedance mismatch.

11 Impedance matching does not require equal wave speed

A subtle but important point is that

(61)

depends on a ratio, while

(62)

depends on a product.

Therefore two media can satisfy

(63)

while still having

(64)

A harmonic wave can cross such an ideal matched interface without reflection even though its wavelength changes because

(65)

The frequency remains fixed by the source while the wavelength adjusts to the new wave speed.

12 Impedance across wave physics

The basic idea of impedance is broader than the string:

impedance relates a wave’s generalized effort variable to its generalized flow variable.

For the string, these variables are transverse force and transverse velocity. In acoustics, a characteristic impedance relates acoustic pressure to particle velocity. In Electromagnetism, wave impedance relates electric and magnetic fields.

The formulas and units differ between physical systems, but the recurring ideas are the same:

• a traveling medium has a characteristic relation between paired wave variables;
• that relation determines how much power a given amplitude transports;
• changes in characteristic impedance cause reflection;
• matching impedances suppresses reflection.

This is one reason the string is such a useful first model for later acoustic, optical, RF, and transmission-line wave physics.

13 Worked Example 1: Compute characteristic impedance

An ideal string has

(66)

Find the wave speed and characteristic impedance.

Solution

The wave speed is

c	=	(67)
	= m/s	(68)
	= 100 m/s.	(69)

The impedance is

Z0	= μc	(70)
	= (0.010)(100) kg/s	(71)
	= 1.00 kg/s.	(72)

Thus

(73)

As a check,

(74)

14 Worked Example 2: Force and instantaneous power

A right-moving wave travels on a string with

(75)

At one instant and position, the transverse velocity is

(76)

Find the transverse force transmitted in the positive x direction and the instantaneous power. Then repeat for a left-moving wave with the same local transverse velocity.

Solution

For the right-moving wave,

(77)

Therefore

(78)

The power is

(79)

Thus

(80)

For a left-moving wave,

(81)

and

(82)

The magnitude of the characteristic impedance is unchanged; the sign of the power identifies the transport direction.

15 Worked Example 3: Average power from impedance

A string has

(83)

A right-moving sinusoid has amplitude

(84)

and frequency

(85)

Find Z₀, the RMS transverse velocity, and the average power.

Solution

First,

(86)

The angular frequency is

(87)

With

(88)

the RMS transverse velocity is

vrms	=	(89)
	= m/s	(90)
	≃ 0.355 m/s.	(91)

Then

⟨P⟩	= Z0vrms2	(92)
	= (1.20)(0.355)2 W	(93)
	≃ 0.152 W.	(94)

Therefore

(95)

(96)

(97)

This is the same numerical result obtained earlier from μA²ω²c.

16 Worked Example 4: Required amplitude for a desired average power

A string has characteristic impedance

(98)

What displacement amplitude is required for a right-moving sinusoidal wave of frequency

(99)

to carry average power

(100)

Solution

Use

(101)

Solve for A:

(102)

The angular frequency is

(103)

Hence

A	= m	(104)
	≃ 2.25 × 10−3 m.	(105)

Thus

(106)

17 Worked Example 5: Reflection from an impedance change

A sinusoidal wave travels from a string with

(107)

into a second string with

(108)

Find the displacement reflection coefficient, displacement transmission coefficient, reflected-power fraction, and transmitted-power fraction.

Solution

The displacement reflection coefficient is

r	=	(109)
	=	(110)
	= −0.60.	(111)

The negative sign means the reflected displacement is inverted.

The displacement transmission coefficient is

t	=	(112)
	=	(113)
	= 0.40.	(114)

The reflected-power fraction is

(115)

The transmitted-power fraction is

𝒯	=	(116)
	=	(117)
	= 0.64.	(118)

Thus

(119)

(120)

The power check is

(121)

18 Worked Example 6: Match impedances while changing wave speed

Medium 1 has

(122)

Medium 2 has tension

(123)

Choose μ₂ so that the two characteristic impedances match. Then find both wave speeds.

Solution

For medium 1,

(124)

For matching,

(125)

Since

(126)

we need

μ2	=	(127)
	= kg/m	(128)
	= 5.625 × 10−3 kg/m.	(129)

Thus

(130)

Now compute the speeds:

c1	=	(131)
	= 90 m/s,	(132)

while

c2	=	(133)
	= 160 m/s.	(134)

Therefore

(135)

The impedances match even though the wave speeds do not. An ideal harmonic wave can therefore have zero reflection while its wavelength changes across the interface.

19 Common mistakes

• Mistake: confusing impedance with wave speed. For a string, c depends on T∕μ, while Z₀ depends on Tμ.
• Mistake: dropping the propagation-direction sign. Z₀ is positive, but the force-velocity relation changes sign between right- and left-moving waves under the chosen positive-x convention.
• Mistake: using Z₀ = F∕u instead of force divided by velocity. Mechanical impedance pairs force with velocity.
• Mistake: applying the one-way relation F = Z₀u_t to an arbitrary Standing Wave. That relation assumes a pure right-moving component.
• Mistake: treating the displacement transmission coefficient t as a power fraction. Power transmission also depends on the impedance ratio.
• Mistake: assuming impedance matching requires equal wave speeds. Equal Z₀ does not imply equal c.

20 What WM22 adds to the wave-mechanics picture

The string-wave sequence now has a compact force-velocity transport relation:

(136)

For one-way waves,

(137)

and

(138)

For a sinusoid,

(139)

At an ideal interface,

(140)

Impedance therefore connects three ideas that were previously introduced separately:

(141)

This framework will later transfer naturally to acoustic and electromagnetic wave impedance. It also prepares the way for characteristic impedance in transmission lines.

References