- Problem Setup
- The Differential Equation
- General Solution
- Satisfying the Initial Conditions
- Examples: plucked string, struck string
- Problems
- References

Consider a flexible string held stationary at both ends and free to vibrate transversely subject only to the restoring forces due to tension in the string. Deduce the DE for such systems and define all parameters that distinguish the systems. Do this with a string of normal length l and using the coordinates depicted here.

Figure showing coordinates and defining symbols for the transverse vibrating string. The length of the string is *l*,
distance along the string from the left end is* x*, its displacement or wave form
is *y(x,t)*.

Our hypothesis is that the string is under constant tension, *T*, established
when it was stretched between its fixed end points. The transverse displacement at
position *x* along the string at time *t* is denoted *y(x,t)*. Our
aim is to deduce the partial differential equation (PDE) governing *y(x,t)*.

The tangent to the string at *y(x,t)* is indicated in the figure; it forms the
hypotenuse of a right triangle with legs parallel and perpendicular to the undisplaced
string position. Also shown is a copy of the tangent triangle (legs *dx*, *dy*,
and hypotenuse *ds*) and a similar triangle with legs *T _{x}* (the
component of tension along the x-direction),

Consider a short segment of the string, length dx, at x where its displacement is *y(x,t)*.
The mass of such a segment is mdx where m is the linear mass density of the string. Denote
the acceleration of this segment as *y"(x,t) = ¶ ^{2}y/¶t^{2}*. According to Newton's laws, the acceleration is
determined by the net force on the segment: m

*¶ ^{2}y/¶t^{2}
= c^{2} ¶^{2}y/¶x^{2}*,

the one-dimensional wave equation (a partial differential equation, second order in
each of *t* and *x*). The boundary conditions are *y(0) = y(l) = 0*, and initial conditions are *y(x,0)* and *y'(x,0)*.

The vibrating string problem can be solved using the method of separation of variables.
Since *y* is a function of *x* and* t*, we look for a solution in the
form of a product, *y(x,t) = X(x)T(t)*. Upon substitution into the wave equation,
the product form requires that *X(d ^{2}T/dt^{2}) = c^{2} T(d^{2}X/dx^{2})*.
Now the variables can be separated if both sides are divided by the Product

Each of these ODEs has general solutions in circular functions: *X(x) = Asin(f x/c)
+ Bcos(f x/c) and T(t) = Csin(ft) + Dcos(ft)*, where *A, B, C*, and *D*
are arbitrary integration constants.

The product form of solution is called a "standing wave" for the following
reasons. For given *f, A*, and *B*, the factor *X(x)* is a wave shape
giving the string displacement at *x*. For the same f and for given *C* and *D*,
the factor *T(t)* oscillates with angular frequency *f* alternating between
positive and negative values. Thus, the wave shape, *X(x)* does not change with
time, only its amplitude oscillates as dictated by *T(t)*.

Fundamental and overtones or harmonics

With both ends of the string tied to rigid walls, the displacements at the boundaries are given by y

(0,t) = y(l,t) = 0. From the left boundary we are led toB(Csin(ft)+Dcos(ft)) = 0. Supposing that theT(t)¹0, it means thatB=0. From the right boundary condition we are led toAsin(fl/c)[Csin(ft)+Dcos(ft)] = 0. Again, the time factor is not zero so that the frequency has to satisfysin(fl/c) = 0orf = npc/lwherenis an integer. Standing wave solutions, those that factor according toX(x)T(t), exist only for "natural", normal, characteristic or eigen- frequencies determined by the string length and wave speed (l, and c). Whenn=1the standing wave is the "fundamental", whenn=2it is the first overtone, and so forth.

General SolutionIn general, the displacement of the vibrating string is given by

y(x,t)= ,

a superposition (linear combination) of many natural modes of vibration of the string. Since the sine and cosine functions are independent, it is always possible to determine the values ofCand_{n}Dfrom the initial wave form y(x,0), and initial wave velocity y'(x,0) using_{n}, respectively. Clearly, this amounts to two applications of the Fourier series to find the Fourier coefficients

Dand_{n}C._{n}

**Plucked string.**A uniform string under tension and of length L is displaced at its center a distance h and released from initial velocity zero. Find its displacements as a function of time y(x,t). That is, the initial velocity is y'(x,0) = 0 while its initial displacement is y(x,0) = 2hx/L for 0<x<L/2 and y(x,0) = 2h(L - x)/L for L/2<x<L. This is a model for plucked strings on a harp or guitar. Answer: y(x,t) = . Note that the odd overtones are present but even overtones are absent; the amplitudes alternate in sign and decrease with the square of the overtone frequency.**Struck string.**In contrast to a plucked string, the struck string is initially not displaced but has initial velocity imparted by a hammer applied at some segment. Suppose a hammer having width s strikes the string at a position d so that the initial string velocity is y'(x,0) = starting from resting position*y(x,0) = 0*. Find its displacements as a function of time*y(x,t)*. Answer:*y(x,t) =*. In this case note that both even and odd overtones are present; the amplitudes are all of the same sign and decrease with the square of the frequency.

- A uniform steel piano string of length 5 feet is under a tension of 900 pounds throughout its length. The wire has linear density 0.027 lb/ft and cross sectional radius of 0.05 in. (a) Calculate the velocity of transverse waves in the string, c. (b) What is the fundamental frequency of vibration of this string?
- A uniform string with length L under tension is plucked at x = L/3 with an amplitude h and released. Find the resulting motion y(x,t).

- Schaum's Outline Series,
*Theory and Problems of Mechanical Vibrations*[Schaum Publishing Co., 1964]. - N. W. McLachlan,
*Theory of Vibrations*[Dover, 1951].

Created or up-dated 08/03/99
by R.D. Poshusta