The "small amplitude" assumption in the derivation of the wave equation for the string

Question

I am reading about the wave equation for transverse waves in a string from the book Mathematics of wave propagation (2000) by J. Davis. On page 10, just before the derivation of the (one-dimensional) wave equation for the string, we read:

It is necessary that the amplitude of the waveform be small, otherwise nonlinear effects take over and the wave shape changes."

(The wave is assumed to be nondispersive).

My first query is: could someone explain what those "nonlinear effects" are and state the form of a nonlinear wave equation PDE in the case of large amplitude waves?

My second question stems from an apparent inconsistency between the derivation of the classical wave equation for the string:

$$\frac{\partial^2u}{\partial t^2}= {v^2}\frac{\partial^2u}{\partial x^2} $$

and the fact that the d´Alembert general solution:

$$u(x, t) = f(x+vt)+g(x-vt)$$

does not require any assumption on the amplitude. Since these are solutions to the wave equation (without any amplitude assumption), why would the derivation of the wave equation insist on the small-amplitude assumption?

The small-amplitude assumption is found in most (but not all) sources on the wave equation on a string. For example, it is in David Morin's online book on Waves (chapter 4, page 2). Is it really necessary?

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Answer 1

A propos "the restoring force being proportional to the displacement," a textbook that does not use the small-amplitude assumption in the derivation of the wave equation for the string is Fourier Analysis: An Introduction, by Elias M. Stein, Rami Shakarchi, pages 6-7. The vibrating string is idealized as a system of a (large) number $N$ of masses linked by massless string segments of equal length:

The assumption is that the tension force on each mass particle is proportional to the vertical (i.e. normal to the $x$-axis) displacement: $$\frac{\tau}{h}{(y_{n+1}-y_n)}$$ where $h=\frac{L}{N}$ and $\tau$ is "a constant equal to the coefficient of tension of the string." (I interpret this $\tau$ as the constant tension in the string in its equilibrium position along the $x$-axis, before it is disturbed).

So, the restoring force is proportional to the vertical displacement $y_{n+1}-y_n$.

After that, the classical wave equation is derived without the small-amplitude assumption. Either this assumption is unnecessary, or the above derivation is not rigorous.

Answer 2

Following the most recent comment, I am going to try and prove that the “vertical displacement force law” assumption implies the small-amplitude assumption. The small-amplitude assumption is equivalent to the small-angle assumption, where I am referring to the angles shown in the picture below, $\theta_n^-$ and $\theta_n^+$. This is the assumption found in many derivations – for example, in David Morin's book Waves (chapter 4, page 2).

I am going to redraw this picture with a simplified notation, which, intentionally, parallels the notation in Morin, (chapter 4, page 2).

Since we are considering a transverse wave, there is no motion in the $x$-direction. This implies that the $x$-component of the net force on the $n$-th mass is equal to zero, which gives us: $${T_1}cos(\theta_1)= T_2{cos(\theta_2)}$$ I believe that $T_1cos(\theta_1)= T_2cos(\theta_2)=\tau$, the tension in the string in the equilibrium position (when it is stretched along the $x$-axis), since no force in the $x$-direction has been applied. But I am going nowhere in my attempt to prove that the “vertical displacement force law” assumption implies the small-amplitude assumption. Could someone help?

Let me also point out that Morin, Chapter 4, page 2 of Waves, considers “small transverse displacements of the string” and uses this assumption to say: “then we can make the approximation that all points in the string move only in the transverse direction. That is, there is no longitudinal motion.”

But aren’t we assuming a transverse wave? “In physics, a transverse wave is a wave that oscillates perpendicularly to the direction of the wave's advance” (Wikipedia). So, it is not necessary to prove that there is no longitudinal motion. Using the small transverse-displacement assumption, and the small-angle assumption, Morin concludes that the tension vector has constant magnitude throughout the string, and says “Let’s call this common tension $T$." This $T$ has better be the tension $\tau$ in the string in the equilibrium position (before it is perturbed), since there is no force is applied along the $x$-direction.

Finally, let me point out that a transverse-displacement assumption is a weaker assumption than "the restoring force is transverse and proportional to the transverse displacement." My impression is that Morin uses no physical assumption at all - just the mathematical assumption of "small transverse displacements." I am not satisfied with the rigor in his proof, and I still would like to see a proof that "the restoring force is transverse and proportional to the transverse displacement" is equivalent to (or at least that it implies) "small transverse displacements" in the string.

Answer 3

OK, I think I finally understand this. The derivation of the wave equation for the string under the assumption that the restoring force is purely transversal, without any small-amplitude assumption, is mathematically correct, but physically unrealistic. The problem is that no real-world string satisfies the assumption that "the restoring force is purely transversal." I will now explain this.

When I said (above) that Morin uses no physical assumption at all, I meant to say that a constitutive relation for the string is needed. The most common constitutive relation is that the string is a perfectly elastic material, which means that the tension (a scalar) in the string is proportional to its (arc) length. Now, a 1959 paper by Joseph Keller in the American Journal of Physics, Large Amplitude Motion of a String, proves the following (page 585. The italic is mine):

The assumption of a perfectly elastic string "is the only stress-strain law for which purely transverse motions of a string are possible when the amplitude of the motion is not assumed to be small."

Unfortunately, a perfectly elastic string cannot possibly exist in the real world, since a string in which the tension is proportional to its (arc) length would necessarily contract to zero length if all tensile forces were removed from it. A "piece of rubber, if stretched to several times its contracted length, satisfies this law approximately" (J. Keller).

I now understand what T. P. Ho said above, that "The "vertical displacement" force law is THE small amplitude approximation." In other words, if you want a purely transverse restoring force in a real-world string, you can only have it for small vibration amplitudes.

A more recent paper that explains the situation is Strings, Chains, and Ropes by Darryl Yong. The author writes, in the Introduction (page 771):

Derivations of the wave equation that make unnecessary assumptions are likely to give students the false impression that deriving PDEs is an ad hoc and unrealistic process. This impression causes students to be less comfortable deriving PDEs on their own and therefore less competent mathematical modelers. Given that the wave equation is the canonical example of a hyperbolic PDE, it is important that students be introduced to the wave equation in a flexible, physically realistic, and systematic way."