String vibration

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Transverse string wave longitudinal motion

ARticle[edit | edit source]

Intro[edit | edit source]

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.[1] This discussion focuses on making a subset of those approximate solutions more accessible to novice lovers of physics.

Three tricks:[edit | edit source]

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.


First derivative revisited[edit | edit source]

Most readers have probably seen a definition of the derivative that can be found at:


Second derivative[edit | edit source]

w:Finite difference coefficient

Let "1" be a small parameter[edit | edit source]


Calculation of force in Figure 1[edit | edit source]


3[edit | edit source]







algebra for 3[edit | edit source]

Extended content

x1: equals minus




y1: equals minus




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.[2] Defining to be the vector associated with this displacement, we have,

We denote

The convention used by OpenStax Physics[3] 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:


Wave equation[edit | edit source]

List of subpages

Leave as exercise for the readers to verify the Table (with both compact and PDE forms). And also to realte X to x.

Equation for kappas and a[edit | edit source]

Define  =>   => 


Allowing [edit | edit source]

Finding the longitudinal component of a transverse wave.

Note that . Let be real, and define,

Therefore,  and , becomes:





Wave-wave interactions[edit | edit source]

Product rule[edit | edit source]

From Phasor_algebra:


problem[edit | edit source]

Variables[edit | edit source]


* position with unit vectors

* coordinate variable parallel to string


* 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


* 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.


* zeta is used for third dimension (polarized waves)



Templates[edit | edit source]

Aligned wikitable[edit | edit source]

Caption text
Header text Header text Header text
Example Example Example
Example Example Example
Example Example Example

Equation[edit | edit source]

We create a four-term expression using (4) and (5). Then we move the terms involving energy density and the Poynting vector to the LHS to obtain:






Key equation[edit | edit source]






Outline[edit | edit source]

  1. Lumped sum model (linear) for transverse and longitudinal waves.
  2. Need image
  3. Calculate field energy density and pose simple question between two choices. One is partially wrong and the other is completely wrong.
  4. See if I can derive the nonlinear interaction obtained by Rowland

Footnotes[edit | edit source]

  • Need to look at: P. M. Morse and K. U. Ingard, Theoretical Acoustics ~McGraw–Hill, New York, 1968 referenced in Rowland 1999.

TRASH[edit | edit source]



2[edit | edit source]

Recall that for small ,

  1. David R Rowland 2011 Eur. J. Phys. 32 1475 Rowland, David R. "The potential energy density in transverse string waves depends critically on longitudinal motion." European journal of physics 32.6 (2011): 1475.
  2. Rowland, et al, use the symbol to describe motion in the other transverse direction.
  3. OpenStax University Physics Volume I Section 5.5: Newton s third law of motion'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.