“Among vibrating bodies there are none that occupy a more prominent position than Stretched Strings. From the earliest times they have been employed for musical purposes ... . To the mathematician they must always possess a peculiar interest as a battle-field on which were fought out the controversies of D’Alembert, Euler, Bernoulli, and Lagrange relating to the nature of the solutions of partial differential equations. To the student of Acoustics they are doubly important." --"Lord Rayleigh:[32, Vol. I, Chap. VI].[1]
https://openstax.org/books/university-physics-volume-1/pages/5-5-newtons-third-law#:~:text=Newton's%20third%20law.-,Newton's%20Third%20Law%20of%20Motion,the%20force%20that%20it%20exerts.&text=F%20%E2%86%92%20AB%20%3D%20%E2%88%92%20F%20%E2%86%92%20BA%20
openstax.org/books/university-physics-volume-1/pages/5-5-newtons-third-law
Transverse string wave lumped mass without text
This discussion was motivated by the fact that the flow of energy associated with Poynting vector does not seem physical. This is not the only case of a conservation law seems to leave unanswered questions about the details. Here we consider the conservation of energy in a stretched string. In this case, there is a unique answer to the question, "where is the potential energy". What is odd is how difficult that question is to answer. Rowland published what seems to be acceptable but approximate solutions for some special cases.
This discussion focuses on making a subset of those approximate solutions more accessible to novice lovers of physics.
It is worth pointing out that stax exchange seems to be wrong on this topic: https://physics.stackexchange.com/questions/414521/does-a-vibrating-string-produce-changes-in-tension-in-the-tring
Figure 1. A mass

is attached to a spring of length

. The mass has been displaced from its equilibrium point by

direction and by

direction. Both

and

are assumed small for a wave in the linear regime. The vector

represents the distance between consecutive lumped masses when the wave amplitude is zero. It is assumed that

because the string is under tension, and

is the length of the relaxed spring of spring constant

. In expressing Newton's second law, the acceleration is written as

using the convention that a dot denotes differentiation with respect to time.
XXX
First derivative revisited[edit | edit source]
Most readers have probably seen a definition of the derivative that can be found at:
- Wikipedia:Simple:Special:Permalink/7230410#Definition_of_a_derivative.
w:Finite difference coefficient
Let "1" be a small parameter[edit | edit source]
- Let

Calculation of force in Figure 1[edit | edit source]
introducing kappa as spring constant[edit | edit source]
Used https://www.symbolab.com/
Constructing the wave equation[edit | edit source]
Figure 2: Free-body force diagram for lumped mass model for vibrating string. We label the three masses shown with the consecutive integers

Here we adopt the convention that
unit of length. We label masses with the variable
that represents each mass by an integer
. To obtain a wave equation we focus on the three consecutive integers,
. When the string is at equilibrium (i.e., zero wave amplitude), we can also use non-integral values of
to form a coordinate system that labels points in space between the masses, as shown in the top of Figure 2.
For non-zero wave amplitude, each mass can move away from its equilibrium point by
in the x direction and
in the y-direction (Rowland, et al, use the symbol
to describe motion in the other transverse direction.)
Defining
to be the vector associated with this displacement, we have,
We denote
The convention used by OpenStax Physics[2] is that
refers to the force on object A by object B. To keep the the notation in Figure 2 compact, we define the displacement vector from A to B as:
<math></math>

Leave as exercise for the readers to verify the Table (with both compact and PDE forms). And also to realte X to x.
Define
=>
=>
UNDER CONSTRUCTION: Allowing ℓ≠1[edit | edit source]
The dimensional analysis conventions introduced in OpenStax University Physics permit us to show that () and () are equivalent to results obtained in reference <sub><big><big>
From from reference RowlandEJP:
Symbolic computation probably renders this exercise unnecessary, but one way to "guess" the wave equation when
is to use dimensional analysis, though if you want certainty it might be better to repeat all the steps with the extra term included. Using
From Phasor_algebra:
<math></math>
* position with unit vectors
* coordinate variable parallel to string
Equilibrium
* spring constant
*relaxed spring length
* equilibrium spring length (no wave present)
* equilibrium tension in string
* linear mass density at equilibrium
*
* transverse wave speed
*longitudinal wave speed
Perturbation
* x-deviation from equilibrium (longitudinal)
* y-deviation from equilibrium (transverse)
* energy density (kinetic+potential) kappa is mine. I don't like their e or k_e.
Avoid
* zeta is used for third dimension (polarized waves)
*
*
- Need to look at: P. M. Morse and K. U. Ingard, Theoretical Acoustics ~McGraw–Hill, New York, 1968 referenced in Rowland 1999.
elsewhere
Temp: {{#lst:String vibration|toc}}
- ↑ David R., and Colin Pask. "The missing wave momentum mystery." American Journal of Physics 67.5 (1999): 378-388.
- ↑ openstax.org/books/university-physics-volume-1/pages/5-5-newtons-third-law