Wave Mechanics: Resonance

WM10 showed that a finite wave system can support standing-wave patterns with fixed nodes and antinodes. Once boundary conditions are imposed, not every wavelength is allowed. Only certain standing-wave patterns satisfy the boundaries, and each allowed pattern has its own natural frequency.

WM11 asks the next question:

What happens when an external periodic driver tries to force the system to oscillate?

The answer is resonance. A periodically driven system responds most strongly when the driving frequency is near one of its natural frequencies. This connection between natural modes, driving, damping, and large response is central to strings, air columns, structures, electrical resonators, optical cavities, and many other wave systems [1, 2, 3, 4].

1 Natural modes come first

Consider a string of length L fixed at both ends. The displacement must vanish at

x = 0 and x = L.

(1)

A convenient standing-wave form is

u (x,t) = B sin(kx )cos(ωt).

(2)

The first boundary condition is automatically satisfied because

sin(0) = 0.

(3)

The second boundary condition requires

sin(kL ) = 0.

(4)

Therefore

kL = nπ, n = 1,2, 3,...

(5)

and the allowed Wavenumbers are

|----------| | nπ- | -kn-=--L-.-|

(6)

Using

2π kn = --, λn

(7)

we obtain

|----------| | 2L- | |λn = n .| -----------

(8)

If the wave speed is v, then WM08 gives

v f = -. λ

(9)

Hence the allowed natural frequencies are

|-----nv---------| |fn = --- = nf1. | ------2L---------|

(10)

The lowest natural frequency is the fundamental frequency,

|---------| f1 = -v-. | -----2L----

(11)

For an ideal uniform string fixed at both ends, the higher natural frequencies are integer multiples of the fundamental and are commonly called harmonics [4].

Figure. The first three normal modes of a string fixed at both ends. Each allowed spatial pattern has its own natural frequency: f₁, 2f₁, and 3f₁.

2 Normal mode is not the same as resonance

These two terms are closely related but describe different ideas.

A normal mode is an allowed pattern in which the system can oscillate freely at one of its natural frequencies. It is a property of the system and its boundary conditions.

Resonance describes the response of the system to an external periodic driver. If the driver frequency f_d is close to a natural frequency f_n, the corresponding mode can respond with a much larger amplitude than it does when the system is driven far from that frequency.

Thus the logical order is

|-----------------------------------------------------------------------------------------| -boundary--conditions--−→---normal-modes---−→---natural-frequencies--−→---resonant--response.--

(12)

3 Why repeated driving can build a large response

A periodic driver supplies a small amount of motion or energy on every cycle. When the timing is favorable, successive pushes reinforce the motion already present. The familiar example is a swing: well-timed small pushes can build a large oscillation.

The same idea applies to wave systems. Suppose a string is driven sinusoidally at one end. A disturbance travels along the string, reflects, and returns. At certain driving frequencies, the repeated forcing is synchronized with an allowed standing-wave pattern. The response then builds strongly.

If the driver is badly detuned from all natural frequencies, successive cycles are not consistently reinforcing. The response is generally much smaller.

Figure. Schematic comparison of resonant and detuned driving. Near a natural frequency, successive pushes reinforce the oscillation coherently; off resonance, the forcing does not remain synchronized with the response.

4 Driving frequency and natural frequency

Let

fd

(13)

denote the driving frequency and let

fn

(14)

denote one natural frequency of the system.

The resonance condition is approximately

|--------| |fd ≈ fn.| ----------

(15)

In angular-frequency language,

ωd-≈--ωn.| ----------

(16)

For a lightly damped system, the strongest response occurs very near the natural frequency. The exact peak can be shifted slightly by damping, depending on which response quantity is being measured. At the level of WM11, the essential physics is that the response becomes large when the drive is near a natural frequency [3, 6].

5 A single mode behaves like a driven oscillator

A useful way to understand resonance is to focus on one normal mode at a time. Its time-dependent amplitude can be modeled like a driven oscillator. A standard one-coordinate model is

m ¨q + b˙q + κq = F0 cos(ωdt),

(17)

where

q(t) is the mode amplitude,
m is an effective modal mass,
κ is an effective restoring coefficient,
b represents damping,
F₀ is the driving-force amplitude,
ω_d is the driving angular frequency.

The undamped natural angular frequency is

|-----∘----| | κ | ω0 = --.| --------m---

(18)

For steady sinusoidal forcing, the response amplitude has the standard form

|--------------------------------| | ----------F0-----------| |Q(ωd ) = ∘ (κ − m ω2)2 + (bω )2 . --------------------d--------d---

(19)

This formula is not derived in WM11; it is included to show mathematically why a response peak appears near the natural frequency. OpenStax and MIT 8.03 develop the driven damped oscillator in detail [3, 6].

6 The resonance curve

If the steady-state response amplitude is plotted against driving frequency, a resonance curve appears.

Figure. Schematic resonance curves for the same natural frequency with different amounts of damping. Less damping produces a taller, narrower peak; more damping produces a lower, broader response.

The plot contains three important ideas.

Far below resonance, the system responds but the amplitude is usually modest.
Near resonance, the response amplitude can become large.
Far above resonance, the system cannot follow the driver efficiently and the response decreases.

Damping changes the resonance dramatically. With weak damping, the peak is high and narrow. With stronger damping, the peak is lower and broader [3].

7 Real resonance peaks are finite

An ideal undamped oscillator driven exactly at its natural frequency is a special mathematical limit. In that limit, continuous coherent driving can make the amplitude grow without bound in the idealized model.

Real systems always contain some combination of damping, energy leakage, material loss, radiation, friction, nonlinear effects, or finite driving time. These effects limit the amplitude.

Therefore, when a real resonance curve is measured, one usually sees a finite peak with a finite width rather than an infinite spike.

8 A wave system can have many resonances

A finite string has many allowed modes, not just one. Each natural mode can therefore produce its own resonant response.

For a fixed-fixed ideal string,

f = nf . n 1

(20)

Sweeping the driving frequency upward can therefore produce a sequence of response peaks near

f1, 2f1, 3f1, ...

(21)

Figure. Schematic frequency sweep of a finite string. Large responses occur near the natural frequencies f₁, f₂, and f₃, with a different standing-wave mode associated with each peak.

This is one of the most important experimental signatures of a bounded wave system: a frequency sweep reveals a spectrum of resonances.

9 Fundamental, harmonics, and overtones

For an ideal string fixed at both ends,

fn = nf1.

(22)

The terminology is:

f₁: fundamental frequency or first harmonic,
f₂ = 2f₁: second harmonic and first overtone,
f₃ = 3f₁: third harmonic and second overtone,
and so on.

The word overtone counts frequencies above the fundamental, whereas the word harmonic counts the fundamental itself as the first harmonic. This distinction prevents a common indexing error.

Not every physical system has natural frequencies that are exact integer multiples of the fundamental. The harmonic relation is a special consequence of the ideal string model and its boundary conditions.

10 Resonance does not create energy

A resonant system can have a large amplitude even when the applied periodic force is comparatively small. This does not violate energy conservation.

The driver performs work on the system repeatedly. Near resonance, the forcing is timed so that energy is transferred efficiently into the oscillation. Damping and other losses remove energy at the same time. In steady state, the average input from the driver balances the average losses.

A quantitative treatment of wave energy, power, and energy flux is deferred to the later energy section of the Wave mechanics series.

11 Worked example 1: find the resonant frequencies of a string

A string of length

L = 1.20m

(23)

supports waves with speed

v = 180m/s.

(24)

The fundamental frequency is

f₁	=	(25)
	= Hz	(26)
	= 75 Hz .	(27)

Therefore

f₂	= 150 Hz ,	(28)
f₃	= 225 Hz ,	(29)
f₄	= 300 Hz .	(30)

If the string is driven near 225 Hz, the third mode is expected to respond strongly.

12 Worked example 2: identify the mode from a driving frequency

A fixed-fixed string has fundamental frequency

f1 = 40 Hz.

(31)

A driver operates at

fd = 160Hz.

(32)

For an ideal string,

fn = nf1.

(33)

Hence

n	=	(34)
	=	(35)
	= 4 .	(36)

The driver is tuned to the fourth harmonic, so the fourth standing-wave mode is expected to be strongly excited.

13 Worked example 3: detuning

Suppose one resonance occurs at

fn = 100Hz.

(37)

Compare two driving frequencies:

fd1 = 99 Hz, fd2 = 70 Hz.

(38)

Their detunings are

\|f_d1 − f_n\|	= 1 Hz,	(39)
\|f_d2 − f_n\|	= 30 Hz.	(40)

All else being equal, the 99 Hz drive lies much closer to the natural frequency and is therefore expected to produce the larger steady response. The exact amplitudes cannot be determined without additional information about damping and coupling strength.

14 Common mistakes

Mistake: using resonance and normal mode as synonyms. A normal mode is an allowed free pattern; resonance is a forced response near a natural frequency.
Mistake: assuming the driving frequency changes the natural frequency. The driver selects how strongly existing modes are excited; the natural frequencies are properties of the system.
Mistake: assuming every frequency produces a standing wave with large amplitude. Large standing-wave response occurs near allowed resonances.
Mistake: assuming the resonance amplitude is infinite in a real system. Damping and other losses limit the response.
Mistake: confusing harmonic number with overtone number. The second harmonic is the first overtone.
Mistake: assuming all systems have f_n = nf₁. That relation holds for the ideal fixed-fixed string but is not universal.

15 What WM11 adds to the wave-mechanics language

WM10 established standing waves. WM11 adds the distinction among three related ideas:

|---------------------------------------------------------------| mode--shape----↔----natural-frequency----↔----resonant-response.--

(41)

For a fixed-fixed ideal string,

|---------| |---------| |--------| | n-π | | 2L-| | nv-| |kn = L , |λn = n , fn = 2L . ---------- ---------- ----------

(42)

An external driver produces a strong response when

|--------| -fd ≈-fn.-

(43)

Damping controls how tall and how broad that resonant response becomes. These ideas prepare the way for modal expansion, Fourier methods, and the study of energy flow in wave systems.

16 References

References

[1] A. P. French, Vibrations and Waves, M.I.T. Introductory Physics Series, W. W. Norton & Company, 1971.

[2] Frank S. Crawford, Jr., Waves, Berkeley Physics Course, Volume 3, McGraw-Hill, 1968.

[3] William Moebs, Samuel J. Ling, and Jeff Sanny, University Physics, Volume 1, OpenStax, 2016, Section 15.6, “Forced Oscillations.”

[4] William Moebs, Samuel J. Ling, and Jeff Sanny, University Physics, Volume 1, OpenStax, 2016, Section 16.6, “Standing Waves and Resonance.”

[5] Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Volume I, Chapter 49, “Modes.”

[6] Massachusetts Institute of Technology, 8.03SC Physics III: Vibrations and Waves, Lecture 3, “Driven Oscillators, Transient Phenomena, Resonance,” Fall 2016, MIT OpenCourseWare.

[7] Massachusetts Institute of Technology, 8.03SC Physics III: Vibrations and Waves, Lecture 9, “Wave Equation, Standing Waves, Fourier Series,” Fall 2016, MIT OpenCourseWare.

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