Wave Mechanics: Deriving the 1D String Wave Equation from Newton’s Second Law

WM13 introduced the derivative structure

(1)

and showed that translating profiles such as F(x − ct) satisfy it. That was a kinematic result: it described a mathematical relation obeyed by a shape-preserving traveling disturbance.

WM14 asks the dynamical question:

(2)

The answer comes directly from Newton’s second law. A curved string has slightly different Tension directions at neighboring points. The difference between those tension directions produces a transverse force. That force is proportional to the local curvature of the string, and Newton’s second law then turns curvature into transverse acceleration. Under the ideal assumptions developed below, this leads to

(3)

or

(4)

Comparison with the standard one-dimensional wave equation gives

(5)

This derivation is standard in treatments of waves on stretched strings and is a canonical example of how a continuum partial differential equation emerges from Newtonian mechanics [1, 2, 3, 5].

1 The physical model

Consider a thin flexible string stretched primarily along the x direction. Let

(6)

denote its transverse displacement from equilibrium.

The derivation uses several assumptions. They are not merely mathematical conveniences; they define the physical model.

• The string is continuous and perfectly flexible, with negligible bending stiffness.
• The string has uniform linear mass density μ.
• The equilibrium tension magnitude T is approximately constant along the string.
• motion is transverse; longitudinal motion is neglected to first order.
• The slope is small:

  (7)

• Damping, gravity, and distributed external transverse forces are neglected.

The most important approximation is the small-slope condition. The displacement itself need not be zero, but neighboring pieces of the string must make only small angles with the equilibrium x direction. For a sinusoidal wave, the small-slope requirement is roughly controlled by the dimensionless quantity Ak.

These assumptions produce the linear string wave equation. If the slopes become large, the tension varies strongly, the string stretches significantly, or bending stiffness matters, additional nonlinear or higher order terms appear.

2 Choose a short string element

Select a small piece of string extending from x to x + Δx.

If μ is the mass per unit equilibrium length, then the mass of this element is approximately

(8)

The element is pulled by the rest of the string at both ends. Because an ideal string can sustain tension but not bending moment, the tension force at each end acts tangent to the local string direction.

Let the local tangent angles be 𝜃_L at the left end and 𝜃_R at the right end.

3 Geometry connects tangent angle to spatial slope

At any point on the string, the slope of the tangent line is

(9)

This relation is geometric and exact for the graph u(x,t) at a fixed time.

The small-slope assumption gives

(10)

so the standard small-angle approximations apply:

(11)

Therefore

(12)

This is the step that linearizes the force law.

A more exact relation would be

(13)

which reduces to sin 𝜃 ≃ u_x when |u_x|≪ 1.

4 Resolve the tension forces

The right-hand tension contributes a transverse component

(14)

while the left-hand tension contributes

(15)

The net transverse force on the element is therefore

(16)

Using the small-slope approximation,

(17)

In coordinate form,

(18)

The horizontal components are

(19)

For small slopes,

(20)

so the horizontal components cancel to first order. This is consistent with the model assumption that the leading motion is transverse and that the tension magnitude can be treated as constant.

5 Curvature produces a transverse force

Rewrite the transverse force as

(21)

As the element becomes arbitrarily short,

(22)

Thus

(23)

This equation contains the central physical idea:

(24)

If the string is locally straight, u_xx = 0, the tension directions balance in the transverse direction and there is no transverse net force from tension. If the string is curved, the two tension vectors do not cancel transversely.

6 Apply Newton’s second law

The transverse acceleration of the material element is

(25)

The element mass is

(26)

Newton’s second law in the transverse direction is

(27)

Substitute the force and mass expressions:

(28)

Cancel the nonzero element length Δx:

(29)

Finally divide by μ:

(30)

This is the one-dimensional linear wave equation for an ideal stretched string.

7 Identify the wave speed

The standard one-dimensional wave equation has the form

(31)

The string equation is

(32)

Therefore

(33)

and

(34)

The positive root is used for the speed magnitude. Direction is carried by the traveling-wave form F(x − ct) or G(x + ct) rather than by assigning a negative value to the speed magnitude.

This result says:

• increasing the tension makes waves travel faster;
• increasing the mass per unit length makes waves travel slower.

More precisely,

(35)

when μ is fixed, and

(36)

when T is fixed.

8 Dimensional check

A correct physical equation must have consistent units.

Tension has units of force:

(37)

Linear mass density has units

(38)

Therefore

	=	(39)
	= .	(40)

Taking the square root gives

(41)

which is the correct dimension for speed.

The differential equation is also dimensionally consistent. If u is a displacement,

(42)

and

(43)

9 The equation expresses a local feedback law

The string equation can be read physically as

(44)

A region that is locally concave upward has

(45)

so the transverse acceleration is upward. A region that is locally concave downward has

(46)

so the transverse acceleration is downward.

At an inflection point,

(47)

so the string has no transverse acceleration from the local tension imbalance at that instant, even though the displacement or velocity at that point may be nonzero.

This is more informative than viewing the wave equation as a purely symbolic relation between second derivatives. The equation is a local Newtonian law: curvature produces force, force produces acceleration, and the resulting motion changes the curvature at neighboring points.

10 Why the equation is linear

The equation

(48)

is linear in the field u. If u₁ and u₂ are solutions for the same constant T and μ, then any linear combination

(49)

is also a solution.

This mathematical linearity is the reason the superposition principle from WM09 works for the ideal string model.

The linearity comes from the modeling assumptions. In particular, the approximation

(50)

replaces the exact geometric force relation by one that is linear in the slope. If the slope is not small, that simplification fails and nonlinear effects can appear.

11 Consistency with a translating disturbance

WM13 showed that a sufficiently smooth right-moving profile

(51)

obeys

(52)

The mechanical derivation now says that the stretched string obeys

(53)

The two are consistent when

(54)

Thus the traveling-wave speed is no longer just a parameter in a chosen function. The mechanical properties of the string determine it.

A sinusoidal wave

(55)

therefore satisfies

(56)

or

(57)

For the ideal string this is a nondispersive relation: all sinusoidal components have the same phase speed ω∕k = c within the model.

12 The role of initial and boundary conditions

The wave equation does not by itself determine one unique motion. WM12 made that point explicit.

For a finite string we still need initial data such as

(58)

and

(59)

plus boundary conditions such as

(60)

for a fixed-fixed string.

The local differential law

(61)

tells us how the interior evolves. The initial and boundary conditions select the particular physical solution.

13 What changes outside the ideal-string assumptions?

The derivation also shows where more complicated models come from.

If the tension varies with position, a more general linearized string equation has the structure

(62)

rather than simply Tu_xx.

If bending stiffness is important, as in a beam or stiff wire, higher spatial derivatives appear. If damping is important, velocity-dependent terms appear. If slopes become large, geometric nonlinearities appear. If external forcing acts along the string, a forcing term appears on the right-hand side.

Thus

(63)

should be understood as the governing equation of a specific idealized physical system, not as a universal equation for every one-dimensional object that can vibrate.

14 Worked example 1: compute wave speed

A string is under tension

(64)

and has linear mass density

(65)

The wave speed is

c	=	(66)
	= m/s	(67)
	= m/s	(68)
	≈ 122 m/s .	(69)

15 Worked example 2: scaling with tension and density

Suppose an ideal string initially has speed c₀.

If the tension is increased by a factor of four while μ stays fixed,

(70)

So quadrupling tension doubles wave speed.

If instead the linear density is increased by a factor of four while T stays fixed,

(71)

So quadrupling linear density halves wave speed.

16 Worked example 3: find the required tension

A string has

(72)

and must support waves at speed

(73)

From

(74)

we obtain

T	= μc2	(75)
	= (0.020)(75)2 N	(76)
	= 112.5 N .	(77)

17 Worked example 4: connect the wave equation to k and ω

Consider

(78)

The Wavenumber and angular frequency are

(79)

Hence

(80)

If the string has

(81)

then the required tension is

T	= μc2	(82)
	= (0.010)(30)2 N	(83)
	= 9.0 N .	(84)

The same conclusion follows from matching the second derivatives:

(85)

Substitution into

(86)

gives

(87)

which is equivalent to

(88)

18 Common mistakes

• Mistake: using displacement itself as the restoring-force variable. For an ideal stretched string, the local tension imbalance is controlled by curvature u_xx, not directly by u.
• Mistake: assuming u_x is the wave speed. It is spatial slope.
• Mistake: assuming u_t is the propagation speed. For string displacement it is the transverse material velocity at fixed x.
• Mistake: forgetting that the mass of the small element is μΔx.
• Mistake: adding the two vertical tension components instead of taking their signed difference.
• Mistake: using c = T∕μ. The correct speed is the square root c = .
• Mistake: treating constant tension as exact for arbitrary large slopes. It is part of the linear ideal-string approximation.
• Mistake: thinking the wave equation alone fixes the motion. Initial and boundary conditions are still required.

19 What WM14 adds to the wave-mechanics language

The earlier lessons built the mathematical structure of a wave. WM14 supplies the mechanical cause for that structure in an ideal string.

The chain is

(89)

Quantitatively,

(90)

and therefore

(91)

This is the first point in the series where the one-dimensional wave equation has been obtained from a physical law rather than merely recognized as a relation satisfied by a chosen traveling waveform.

20 References

References