## Ideal Vibrating String

The ideal vibrating string is depicted in Fig.6.1. It is assumed to be perfectly flexible and elastic. Once ``plucked,'' it will vibrate forever in one plane as an energy conserving system. The mathematical theory of string vibration is considered in §B.6 and Appendix C. For present purposes, we only need some basic definitions and results.

### Wave Equation

The *wave equation*
for the ideal vibrating string may be written as

where we define the following notation:

As discussed in Chapter 1, the wave equation in this form can be interpreted as a statement of Newton's second law,

*i.e.*, . See Appendix B for a review of basic physical concepts. The wave equation is derived in some detail in §B.6.

### Wave Equation Applications

The ideal-string wave equation applies to any perfectly elastic medium
which is displaced along one dimension. For example, the air column
of a clarinet or organ pipe can be modeled using the one-dimensional
wave equation by substituting air-pressure deviation for string
displacement, and longitudinal volume velocity for transverse string
velocity. We refer to the general class of such media as
*one-dimensional waveguides.* Extensions to two and three
dimensions (and more, for the mathematically curious), are also
possible (see §C.14)
[518,520,55].

For a physical string model, at least three coupled waveguide models
should be considered. Two correspond to transverse-wave vibrations
in the horizontal and vertical planes (two
*polarizations* of
planar vibration); the third corresponds to *longitudinal
waves*. For bowed strings, *torsional waves* should also be
considered, since they affect bow-string dynamics
[308,421]. In the piano, for key ranges in
which the hammer strikes three strings simultaneously, *nine*
coupled waveguides are required per key for a complete simulation
(not including torsional waves); however, in a practical,
high-quality, virtual piano, one waveguide per coupled string
(modeling only the vertical, transverse plane) suffices quite well
[42,43]. It is difficult to get by with fewer
than the correct number of strings, however, because their detuning
determines the entire amplitude envelope as well as beating and
aftersound effects
[543].

### Traveling-Wave Solution

It can be readily checked (see §C.3 for details) that the lossless 1D wave equation

*any string shape*which travels to the left or right with speed

Note that we have and (derived in §C.3.1) showing that the wave equation is satisfied for all traveling wave shapes and . However, the derivation of the wave equation itself assumes the string slope is much less than at all times and positions (see §B.6). An important point to note is that a function of two variables is replaced by two functions of a single (time) variable. This leads to great reductions in computational complexity, as we will see. The traveling-wave solution of the wave equation was first published by d'Alembert in 1747 [100]

^{7.1}

### Sampled Traveling-Wave Solution

As discussed in more detail in Appendix C, the continuous traveling-wave
solution to the wave equation given in Eq.(6.2) can be *sampled*
to yield

where denotes the time sampling interval in seconds, denotes the spatial sampling interval in meters, and and are defined for notational convenience.

### Wave Impedance

A concept of high practical utility is that of *wave impedance*,
defined for vibrating strings as force divided by velocity. As
derived in §C.7.2, the relevant *force* quantity in this case
is minus the string tension times the string slope:

(7.4) |

Physically, this can be regarded as the transverse force

*acting to the right*on the string in the vertical direction. (Only transverse vibration is being considered.) In other words, the vertical component of a negative string slope pulls ``up'' on the segment of string to the right, and ``up'' is the positive direction for displacement, velocity, and now force. The traveling-wave decomposition of the force into

*force waves*is thus given by (see §C.7.2 for a more detailed derivation)

^{7.2}

where we have defined the new notation for transverse velocity, and

*wave impedance*of the string for transverse waves. It is always real and positive for the ideal string. Three expressions for the wave impedance are

The wave impedance simply relates force and velocity traveling waves:

These relations may be called

*Ohm's law for traveling waves*. Thus, in a traveling wave, force is always

*in phase*with velocity (considering the minus sign in the left-going case to be associated with the direction of travel rather than a degrees phase shift between force and velocity).

The results of this section are derived in more detail in Appendix C. However, all we need in practice for now are the important Ohm's law relations for traveling-wave components given in Eq.(6.6).

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Ideal Acoustic Tube

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The Leslie