Wave Mechanics: Boundary Conditions

WM10 and WM11 showed that finite systems support standing waves, normal modes, and resonances. Those results depended on statements such as “the string is fixed at both ends.” WM12 makes that statement mathematical.

A wave equation, or any other differential equation for a field, does not by itself determine one unique physical motion. The equation describes which local behaviors are allowed. Additional information is required to determine which solution is physically realized. Two kinds of information are especially important:

initial conditions, which specify the state of the system at an initial time, and
boundary conditions, which specify what the field must do at the edges or interfaces of the spatial region.

For waves on a string, the most important elementary boundary conditions are a fixed end, where the displacement must vanish, and a free end, where the transverse force must vanish. These two cases produce different reflections and different families of standing-wave modes [1, 2, 3, 5, 6].

1 Initial conditions are not boundary conditions

Suppose a string occupies the interval

0 ≤ x ≤ L.

(1)

The field

u(x,t)

(2)

specifies the transverse displacement of the string.

An initial condition tells us something about the entire string at one instant, for example

u (x,0) = f(x).

(3)

A second initial condition is usually needed to specify the initial velocity,

∂u ---(x,0) = g(x). ∂t

(4)

By contrast, a boundary condition tells us what happens at a particular spatial edge for all relevant times. Examples are

u(0,t) = 0

(5)

∂u- ∂x (L, t) = 0.

(6)

The first fixes the displacement at the left end. The second fixes the spatial slope at the right end. The partial-derivative notation simply means that the slope is measured with respect to x while time is held fixed. A later article will develop partial derivatives more systematically.

Figure. Initial conditions specify the field along an initial-time line, whereas boundary conditions constrain the field along the spatial edges of the domain. A complete wave problem generally requires both kinds of information.

This distinction is fundamental in mathematical physics. A differential equation plus initial and boundary data defines an initial-boundary-value problem.

2 Fixed-end boundary condition

Imagine a string attached rigidly to a wall at

x = 0.

(7)

The endpoint cannot move transversely, so

|-----------| u (0, t) = 0. | ------------

(8)

This is commonly called a fixed or Dirichlet boundary condition. The name Dirichlet refers to the mathematical statement that the value of the field itself is prescribed at the boundary.

For a fixed end, an incident pulse must reflect in such a way that the incident and reflected displacements always cancel at the wall. Feynman describes this by superposing the incoming pulse with an oppositely signed reflected pulse so that the endpoint remains at zero displacement [5].

For a sinusoidal wave incident on a fixed boundary, write

ui(x, t) = A cos(kx + ωt)

(9)

for a wave traveling toward decreasing x. Let the reflected wave be

ur(x,t) = Ar cos(kx − ωt).

(10)

At the boundary,

ui(0,t) + ur(0,t) = 0.

(11)

Therefore

A cos(ωt ) + Ar cos(ωt) = 0,

(12)

which requires

Ar = − A.

(13)

Thus the reflected displacement has the opposite sign:

|---------------------------| ur(x,-t)-=-−-A-cos(kx-−-ωt-).--

(14)

The reflection therefore contains a phase change of π in the displacement. A crest returns as a trough and a trough returns as a crest [3, 5].

Figure. Reflection from a fixed boundary. The endpoint must remain at zero displacement, so the reflected pulse is inverted. The incident and reflected displacements cancel at the wall at every instant.

3 Free-end boundary condition

Now suppose the string terminates in a Light ring that can slide without friction on a vertical support. The endpoint is free to move transversely. The correct boundary condition is no longer zero displacement.

Instead, the transverse force at the end must vanish. For a string under constant Tension T, a small slope produces a transverse component of tension proportional to the spatial slope of the string. At a truly free end there is no external transverse force available to balance a nonzero end slope. Therefore

|-------------| ∂u- | ∂x (L, t) = 0. | ---------------

(15)

This is a free or Neumann boundary condition. A Neumann condition prescribes a derivative of the field rather than the field value itself. MIT 8.03 presents the same zero-slope condition for a massless ring on a frictionless support [7].

For a free end, the reflected pulse is not inverted. The reflected displacement returns with the same sign as the incident displacement. A crest returns as a crest and a trough returns as a trough [3].

Figure. Reflection from a free boundary. The endpoint is allowed to move, but its spatial slope must vanish. The reflected pulse is not inverted, so a crest returns as a crest.

The two elementary reflection rules are therefore

|----------------------------------------------| |fixed end: displacement reflection changes sign | ------------------------------------------------

(16)

and

|----------------------------------------------------| free-end:-displacement--reflection-keeps-the-same-sign.-

(17)

4 Why the free end requires zero slope

The zero-slope condition deserves a physical interpretation.

At an interior point of a slightly curved string, the tension forces from the left and right need not cancel in the transverse direction. Their imbalance accelerates the string element.

At a free endpoint there is only one string segment pulling on the endpoint. If that segment meets the endpoint with a nonzero transverse slope, its tension has a transverse component. A massless freely sliding endpoint cannot sustain a finite unbalanced transverse force. In the ideal limit, the string must therefore meet the support horizontally:

∂u-= 0. ∂x

(18)

This is the mechanical origin of the Neumann condition for the ideal free end.

5 Two fixed ends select integer half-wavelengths

Consider a string occupying

0 ≤ x ≤ L

(19)

with both ends fixed:

u (0, t) = 0, u(L, t) = 0.

(20)

A convenient standing-wave form is

u (x,t) = B sin(kx )cos(ωt).

(21)

The left condition is automatically satisfied. The right condition requires

sin(kL ) = 0.

(22)

Therefore

kL = nπ, n = 1,2, 3,...

(23)

and hence

|----------| | nπ- | |kn = L . | -----------

(24)

Using k = 2π∕λ gives

|----------| |λn = 2L-.| -------n---|

(25)

The fixed-fixed boundary conditions force the string length to contain an integer number of half wavelengths:

|--------| | λ-| L = n 2.| ----------

(26)

This is the mode family used in WM10 and WM11.

6 One fixed end and one free end select odd quarter-wavelengths

Now impose a different pair of boundary conditions:

u(0,t) = 0

(27)

and

∂u-(L, t) = 0. ∂x

(28)

Again choose

u (x,t) = B sin(kx )cos(ωt).

(29)

The fixed condition at x = 0 is satisfied automatically. Differentiate with respect to x:

∂u --- = Bk cos(kx)cos(ωt). ∂x

(30)

At the free end,

cos(kL ) = 0.

(31)

Thus

kL = (2n-−-1-)π-, n = 1,2,3,... 2

(32)

|----------------| |k = (2n-−-1)π-.| --n-------2L------

(33)

The allowed wavelengths are

|-------------| | 4L | λn = ------. | ------2n-−-1---

(34)

If the wave speed is v, then

|----------------| | (2n − 1)v | |fn = ----------.| ----------4L-----

(35)

Only odd multiples of the fundamental appear:

f1, 3f1, 5f1,...

(36)

This family is characteristic of a system with one displacement node and one slope-zero end. The same quarter-wave geometry appears in other wave systems, including idealized air columns with one closed end and one open end, although the precise acoustic boundary variables differ from string displacement [1, 4].

Figure. Boundary conditions select different mode families. A fixed-fixed string has nodes at both ends and permits integer half-wavelengths. A fixed-free string has a node at the fixed end and a slope-zero antinode at the free end, producing odd quarter-wavelength modes.

7 Driven boundaries are nonhomogeneous boundary conditions

A boundary does not have to be fixed or free. It can be prescribed to move. For example, suppose the left end of a string is driven sinusoidally:

|--------------------| |u(0,t) = a cos(ω t). | -----------------d---

(37)

This is a prescribed-displacement boundary condition. Because the prescribed value is not zero, it is called a nonhomogeneous boundary condition.

By contrast,

u(0,t) = 0

(38)

and

∂u-(L, t) = 0 ∂x

(39)

are homogeneous conditions because their right-hand sides are zero.

Driven boundaries provide a natural link to resonance. If the drive frequency is close to one of the natural frequencies allowed by the other boundaries, the corresponding mode can be strongly excited.

8 Boundary conditions at an interface

Not every boundary is an endpoint. Two different media may meet at an interface. In that situation, the field generally does not simply terminate. Instead, incident, reflected, and transmitted waves must satisfy matching conditions at the interface.

The exact conditions depend on the physical field. For a stretched string with no break in the string, the displacement must remain continuous. Force balance supplies a second condition involving tension and spatial slope. For electromagnetic waves, Maxwell’s equations impose continuity conditions on appropriate electric and magnetic field components. Feynman emphasizes that boundary conditions are the rules that make the solutions on the two sides of an interface fit together consistently [8].

WM12 does not derive reflection and transmission coefficients at a material interface. That calculation is reserved for a later article. The important idea here is that an interface problem requires enough matching conditions to connect the solutions on the two sides.

9 Dirichlet, Neumann, and mixed conditions

The two most common mathematical names are:

Dirichlet condition: prescribe the field value, such as
$u (0, t) = 0.$ (40)
Neumann condition: prescribe the normal derivative or slope, such as
$∂u ---(L, t) = 0. ∂x$ (41)

More complicated boundaries can combine field value and derivative. A schematic mixed condition might have the form

au + b ∂u-= h(t). ∂x

(42)

Such conditions can model elastic supports, impedance-like terminations, and other intermediate cases. The physical coefficients determine how strongly the boundary behaves like a fixed, free, or partially transmitting termination.

At this stage, the important point is not to memorize all possible forms. It is to recognize that the physics at the edge is encoded mathematically through a condition on the field and possibly its derivatives.

10 Boundary conditions select the allowed solutions

A useful way to summarize the role of boundary conditions is

|---------------------------------------------------------------| wave law + initial data + boundary data −→ physical solution. | ----------------------------------------------------------------

(43)

The local wave law determines what kinds of behavior are possible in the interior. Initial conditions determine how the motion starts. Boundary conditions determine how the field must behave at the edges. Together they select one physical evolution from the much larger family of mathematically possible waveforms.

This is why changing only one endpoint can reorganize the entire normal-mode spectrum of a finite system.

11 Worked example 1: reflection from a fixed end

An incident displacement pulse reaches a rigid boundary with peak amplitude

+4.0 mm.

(44)

At a fixed end,

u (0, t) = 0.

(45)

The reflected displacement must cancel the incident displacement at the wall. Therefore the reflected pulse has peak amplitude

|---------| -−-4.0-mm--.

(46)

The negative sign indicates inversion of the displacement pulse.

12 Worked example 2: reflection from a free end

An incident crest with peak displacement

+3.0 mm

(47)

reaches an ideal free end.

For the free-end condition,

∂u- ∂x = 0

(48)

at the endpoint. The reflected displacement is not inverted. The reflected pulse therefore has peak displacement

|---------| |+3.0 mm . -----------

(49)

13 Worked example 3: fixed-free normal frequencies

A string of length

L = 0.80m

(50)

has one fixed end and one ideal free end. The wave speed is

v = 96 m/s.

(51)

The allowed frequencies are

fn = (2n-−-1)v-. 4L

(52)

The fundamental is

f₁	= Hz	(53)
	= Hz	(54)
	= 30 Hz .	(55)

The next two allowed frequencies are

f₂	= 3f₁ = 90 Hz ,	(56)
f₃	= 5f₁ = 150 Hz .	(57)

Notice that 60 Hz is not an allowed normal frequency in this ideal fixed-free system.

14 Worked example 4: identify the boundary type

Suppose a standing-wave profile satisfies

u(0,t) = 0

(58)

and at the opposite end

∂u u(L, t) ⁄= 0, ---(L, t) = 0. ∂x

(59)

The left end is a displacement node, so it behaves like a fixed boundary. The right end has zero spatial slope but nonzero displacement, so it behaves like a free boundary.

The system is therefore a

|------------------| -fixed--free-system.-|

(60)

15 Common mistakes

Mistake: treating initial conditions and boundary conditions as the same thing. Initial conditions specify the state at an initial time; boundary conditions specify behavior at spatial edges.
Mistake: assuming a fixed end means zero slope. A fixed end requires zero displacement; its slope need not be zero.
Mistake: assuming a free end means zero displacement. A free end can move; the ideal string condition is zero spatial slope.
Mistake: forgetting the phase inversion at a fixed-end reflection. The reflected displacement changes sign.
Mistake: assuming a free-end reflection is inverted. In the ideal string model, the reflected displacement keeps the same sign.
Mistake: using the fixed-fixed mode formula for a fixed-free system. The boundary conditions determine the allowed Wavenumbers and must be applied before choosing a mode formula.
Mistake: assuming every physical field uses the same boundary variable. Displacement, pressure, electric field, magnetic field, and quantum wavefunctions have different physical matching rules.

16 What WM12 adds to the wave-mechanics language

WM10 and WM11 used boundary conditions implicitly to obtain standing waves and resonances. WM12 makes the boundary conditions themselves explicit.

For an ideal string,

|----------------| |fixed end: u = 0 | ------------------

(61)

and

|------------------| | ∂u | |free end: ---= 0. | -----------∂x------

(62)

These conditions determine both reflection behavior and the allowed mode spectrum. In particular,

|--------------------------------| | nπ- nv-| |fixed-fixed: kn = L , fn = 2L | ---------------------------------

(63)

whereas

|--------------------------------------------| | (2n − 1)π (2n − 1)v | fixed -free: kn = ----------, fn = ---------.| --------------------2L----------------4L------

(64)

The deeper lesson is that a wave equation describes local physics, while boundary conditions encode the physical constraints at the edges. Changing the boundary changes the allowed global solutions.

17 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 16.5, “Interference of Waves,” especially the discussion of reflection at fixed and free boundaries.

[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,” especially Section 49–1 on reflection of waves from a clamped boundary.

[6] Massachusetts Institute of Technology, 8.03SC Physics III: Vibrations and Waves - The Physics of Waves, Fall 2016, MIT OpenCourseWare, Sections 5.1.2, 5.3.2, 5.4, and 5.5 on boundary conditions, fixed ends, free ends, and forced boundary conditions.

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

[8] Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Volume II, Chapter 33, “Reflection from Surfaces,” especially the discussion of boundary conditions and matching electromagnetic fields at an interface.

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