Continuous Strings, Standing Waves, and Normal Modes

Continuous Strings, Standing Waves, and Normal Modes

A long chain of coupled oscillators approaches a continuous medium when the lattice spacing becomes small compared with the wavelength.

The simplest continuous wave-bearing mechanical system is a stretched string.

Let

• T be the approximately constant string Tension,
• μ be the mass per unit length,
• y(x,t) be a small transverse displacement.

For small slopes, the transverse motion obeys

(1)

where

(2)

This is the one-dimensional wave equation.

A string of length

(3)

fixed at both ends cannot support arbitrary single-frequency shapes.

The boundary conditions select a discrete family of normal modes:

(4)

with

(5)

Thus a continuous system has infinitely many normal modes.

The finite-dimensional modal ideas from M06-04 and M06-05 become a Fourier-series description of a continuous displacement field.

1 Small transverse displacement model

Consider a string whose equilibrium position lies along the

(6)

axis.

Let

(7)

be its transverse displacement.

Assume:

• the displacement is small,
• the slope satisfies |∂y∕∂x|≪ 1,
• the tension magnitude is nearly constant,
• longitudinal motion is neglected,
• the string is perfectly flexible.

These assumptions produce the linear string model.

2 Force balance on a small element

Take an element extending from

(8)

to

(9)

Its mass is

(10)

The tension at each end is tangent to the string.

Let the local tangent angle be

(11)

For small slope,

(12)

The net transverse force is therefore

dFy	= T	(13)
	≈ Tdx.	(14)

Newton’s second law gives

(15)

Canceling

(16)

gives

(17)

3 Wave speed

Divide by

(18)

Then

(19)

Define

(20)

Thus

(21)

The dimensions are

(22)

Larger tension increases the wave speed.

Larger linear mass density decreases it.

4 Traveling-wave solutions

A function of the form

(23)

satisfies the wave equation.

It represents a shape traveling to the right at speed

(24)

Similarly,

(25)

travels to the left.

The general d’Alembert solution on an unbounded string is

(26)

Standing Waves arise when oppositely traveling waves of the same frequency interfere.

5 A sinusoidal traveling wave

Consider

(27)

Substitution into the wave equation gives

(28)

Therefore

(29)

for positive

(30)

The ideal string is nondispersive:

(31)

and

(32)

Unlike the discrete chain of M06-06, the ideal continuous string has a linear dispersion relation.

6 Standing wave from two traveling waves

Take equal-amplitude waves traveling in opposite directions:

(33)

and

(34)

Their sum is

(35)

Using a trigonometric identity,

(36)

The spatial shape

(37)

does not translate.

Its amplitude oscillates in time.

This is a standing wave.

7 Fixed-end boundary conditions

Now consider a string extending from

(38)

to

(39)

If both ends are fixed,

(40)

and

(41)

These boundary conditions select the allowed standing-wave patterns.

8 Separation of variables

Seek a product solution

(42)

Substitution into

(43)

gives

(44)

Divide by

(45)

(46)

The left side depends only on time.

The right side depends only on position.

Therefore both must equal a constant.

For oscillatory normal modes, write that constant as

(47)

Then

(48)

and

(49)

where

(50)

9 Solving the spatial equation

The spatial solution is

(51)

The boundary condition

(52)

requires

(53)

Thus

(54)

The second boundary condition,

(55)

requires

(56)

Therefore

(57)

10 Normal frequencies

Since

(58)

(59)

The ordinary frequencies are

(60)

Therefore

(61)

with

(62)

The fixed-fixed ideal string has an exact harmonic spectrum.

11 Normal-mode shapes

Ignoring an arbitrary normalization constant,

(63)

The first few modes are:

X1(x)	= sin ,	(64)
X2(x)	= sin ,	(65)
X3(x)	= sin .	(66)

12 Allowed wavelengths

For

(67)

and

(68)

(69)

Thus the string length contains

(70)

half-wavelengths:

(71)

13 Nodes and antinodes

A node is a point whose displacement remains zero throughout the normal-mode motion.

For mode

(72)

nodes satisfy

(73)

Therefore

(74)

The endpoints are always nodes.

Antinodes occur where the magnitude of the sine is one:

(75)

for the appropriate integer values of

(76)

14 Mode orthogonality

The fixed-end mode functions satisfy

(77)

for

(78)

For

(79)

(80)

Combining both cases,

(81)

This is the continuum analogue of eigenvector orthogonality.

15 General modal expansion

Because the modes form a complete basis for suitable fixed-end shapes,

(82)

Substituting into the wave equation gives, mode by mode,

(83)

Thus

(84)

Therefore

y(x,t)	= ∑ n=1∞	(85)
	× sin .	(86)

A continuous string is therefore an infinite collection of independent harmonic normal coordinates.

16 Initial displacement determines A_n

Suppose

(87)

At

(88)

(89)

Multiply by

(90)

and integrate.

Orthogonality gives

(91)

These are Fourier sine coefficients.

17 Initial velocity determines B_n

Suppose

(92)

Differentiate the modal expansion:

(93)

Project using orthogonality:

(94)

The complete initial state

(95)

therefore determines every modal coordinate.

This is the continuous-field analogue of projecting finite-dimensional initial position and velocity onto Normal eigenvectors.

18 Example 1: fundamental frequency of a string

Let

L	= 0.650 m,	(96)
T	= 120 N,	(97)
μ	= 5.00 × 10−3 kg∕m.	(98)

The wave speed is

c	=	(99)
	=	(100)
	≈ 154.9 m∕s.	(101)

The fundamental frequency is

f1	=	(102)
	=	(103)
	≈ 119.2 Hz.	(104)

The next two harmonics are

(105)

19 Example 2: string plucked into a triangular shape

Suppose the string is displaced to height

(106)

at

(107)

with

(108)

and is otherwise straight between the endpoints and the pluck point.

Thus

(109)

Assume the string is released from rest:

(110)

Therefore

(111)

Evaluating the Fourier integral gives

(112)

The motion is

y(x,t)	= ∑ n=1∞A n cos(ωnt)	(113)
	× sin .	(114)

The pluck position controls which harmonics are excited.

20 Midpoint pluck

For a midpoint pluck,

(115)

Then

(116)

For even

(117)

(118)

Therefore

(119)

Only odd harmonics are excited by a perfectly symmetric midpoint pluck.

For the first few nonzero terms,

(120)

(121)

and

(122)

The

(123)

decay reflects the relatively smooth, piecewise-linear initial shape.

21 Nodes suppress selected modes

Suppose the string is plucked at a position

(124)

A mode whose node lies exactly at

(125)

can have zero overlap with certain symmetric pluck profiles.

For the triangular result,

(126)

Thus if

(127)

that harmonic is absent.

The geometry of the initial condition therefore selects the modal spectrum.

22 Kinetic energy of a string

The kinetic energy of an element

(128)

is

(129)

Thus

(130)

23 Elastic potential energy

For small slope, the extra length of a string element is approximately

(131)

Work against the tension gives

(132)

Therefore total energy is

(133)

24 Energy in modal coordinates

Insert

(134)

Orthogonality eliminates cross terms.

The result is

(135)

where

(136)

Thus each string normal mode behaves as an independent harmonic oscillator.

25 Effective modal mass

The modal energy can be written

(137)

Comparing with the previous expression gives

(138)

for the unnormalized choice

(139)

The numerical modal mass depends on the normalization convention for

(140)

The physical field

(141)

does not.

26 Continuum orthogonality as mass orthogonality

For a nonuniform string with position-dependent density

(142)

the natural modal inner product contains a Weight:

(143)

This is the continuous analogue of the finite-dimensional mass inner product

(144)

For a uniform string,

(145)

is constant and can be factored out, leaving the ordinary sine orthogonality integral.

27 Fixed-free string

Now suppose

(146)

is fixed but

(147)

is free in the idealized transverse sense.

The fixed end requires

(148)

A free end has zero transverse force.

For small slope this gives

(149)

With

(150)

the free-end condition requires

(151)

Thus

(152)

Therefore

(153)

Only odd multiples of the fundamental occur.

28 Free-free string and the zero mode

For ideal free ends,

(154)

There is a constant mode

(155)

with

(156)

It corresponds to uniform transverse translation in the idealized mathematical model.

The remaining modes have

(157)

Boundary conditions determine the spectrum just as constraints determine normal modes in finite-dimensional mechanics.

29 Discrete chain to continuous string

M06-06 derived the discrete chain equation

(158)

For slow spatial variation,

(159)

Thus

(160)

The string equation has

(161)

Both have the same continuum form:

(162)

30 Why the continuous spectrum is still discrete on a finite string

The string has infinitely many degrees of freedom because

(163)

is defined at continuously many positions.

Yet a finite string with boundary conditions has discrete allowed

(164)

The reason is not spatial discreteness.

It is the boundary-value eigenproblem.

Only certain spatial functions satisfy both the differential equation and the endpoint conditions.

Thus:

• discrete lattices can have discrete or nearly continuous allowed k values depending on size and boundary conditions,
• continuous media can have discrete normal frequencies when confined to finite domains.

31 Normal modes versus traveling waves

A normal mode of a finite fixed string is naturally represented as a standing wave.

A traveling wave is more natural on an unbounded string or before boundary reflections are combined.

The two descriptions are related.

A standing mode can be decomposed into equal-frequency left- and right-traveling waves.

Conversely, repeated reflection of traveling waves from boundaries builds the standing-wave normal-mode structure.

32 Common mistakes

1. Using mass density per volume instead of linear mass density μ for the ideal string.
2. Forgetting the small-slope approximation in the derivation.
3. Treating the tension’s horizontal and transverse components as independently constant.
4. Writing c = T∕μ instead of c = .
5. Confusing traveling-wave speed with transverse material-point speed.
6. Assuming a standing-wave pattern propagates along the string.
7. Forgetting that fixed boundaries require zero displacement.
8. Using k_n = 2nπ∕L instead of nπ∕L for fixed-fixed modes.
9. Confusing angular frequency ω_n with ordinary frequency f_n.
10. Forgetting the factor of 2 in f_n = nc∕(2L).
11. Calling every Fourier component a distinct physical string rather than a mode of one string.
12. Projecting initial displacement but forgetting initial velocity.
13. Assuming all pluck positions excite all harmonics.
14. Forgetting that modal mass depends on mode normalization.
15. Applying fixed-fixed boundary conditions to fixed-free or free-free problems.

33 Practice exercises

1. Derive the transverse force on a short string element and obtain the wave equation.
2. Verify dimensionally that c = has units of speed.
3. Show directly that f(x − ct) satisfies the wave equation.
4. Derive the standing-wave form from two counter-propagating sinusoidal waves.
5. Use separation of variables to derive the fixed-fixed spatial and temporal equations.
6. Derive k_n = nπ∕L and f_n = nc∕(2L).
7. Find the nodes and antinodes of mode n.
8. Prove sine-mode orthogonality over 0 ≤ x ≤ L.
9. Derive the modal coefficient A_n from an arbitrary initial displacement.
10. Derive the modal coefficient B_n from an arbitrary initial velocity.
11. Derive the triangular-pluck coefficients.
12. Show that a midpoint pluck excites only odd harmonics.
13. Derive the total string energy and show that modal cross terms vanish.
14. Derive the fixed-free spectrum.
15. Explain how the discrete-chain continuum limit and the finite-string normal modes fit into one common normal-mode framework.

34 Summary

For a uniform stretched string,

(165)

or

(166)

A fixed-fixed string has normal modes

(167)

with

(168)

and

(169)

An arbitrary fixed-end motion is

y(x,t)	= ∑ n=1∞	(170)
	× sin ,	(171)

where

(172)

and

(173)

The continuum field therefore decomposes into infinitely many independent harmonic normal coordinates.

This extends the normal-mode machinery from discrete mechanics directly into wave physics.

References

References