Arrange by : M.Khalis

If a portion of a stretched string is displaced from its equilibrium position and released. Then;

We observe that the displacement does not remain fixed in its initial position, but breaks up into two separate disturbances that propagate along the string.

One moving to the right and the other to the left with equal speed .

Propagation of a transverse disturbance along a stretched string. (a) Initial stationary disturbance at t = 0. (b) Disturbances moving to right and left at tl > 0. (c) Disturbances moving to right and left at t2 > tl.

• We consider the forces that tend to return the string to its equilibrium position, and try to derive a wave equation.

• Solutions of this wave equation satisfying the appropriate initial and boundary conditions will completely define the motion of the string.

• Assume a string of uniform linear density pL and negligible stiffness, stretched to a tension T great enough that the effects of gravity can be neglected.

& • that there are no dissipative forces (such as those

associated with friction or with the radiation of acoustic energy).

• Next fig. shows an infinitesimal element of the string with equilibrium position x and equilibrium length dx.

An important wave motion is the propagation of longitudinal (compressional) waves, often encountered in solid bars (then at low frequencies, in gas-filled tubes and ducts with rigid walls).

As a longitudinal disturbance moves along a bar, the displacements of particles of the bar are essentially parallel to its axis.

The lateral dimensions of the bar are small compared with its length, each cross-sectional plane of the bar may be considered to move as a unit. (Actually the bar shrinks somewhat laterally as it expands longitudinally, but for thin bars this lateral motion may be neglected.)

• A number of acoustic devices utilize longitudinal vibrations in bars.

• Frequency standards used for producing sounds of definite pitches can be constructed from rods of various lengths.

• When longitudinal vibrations are excited in such rods, the frequency of vibration is observed to be inversely proportional to the length of the rod (if all are of the same composition).

• Longitudinal vibrations in nickel tubes are often used to drive the vibrating diaphragm of a sonar transducer.

• Piezoelectric crystals may be cut so that the frequency of longitudinal vibration in a selected direction in the crystal is used either to control the frequency of an oscillatory electric current or to drive an electroacoustic transducer.

Important to understand acoustic wave. Mathematic expression of transmission

acoustic plane waves of fluid is similar with transmission of compressional waves along a bar.

If fluid confined as rigid pipe, there is close analogy between the boundary condition.

x X + dx

𝜉 + d

Figure 2 ; Longitudinal strain d / dx of an element length dx in a 𝜉bar.

Let the coordinates of the left and right ends of the bar be x = 0 and x = L, and

Consider a short segment dx of the unstrained bar lying between x and x + dx.

Assume that the forces cause the plane originally located at x to move a distance 𝜉to the right, and the plane originally located at x + dx to move a distance + d to 𝜉 𝜉the right (Fig. 2).

• Whenever a bar is strained, elastic forces are produced. These forces act across each cross-sectional plane in the bar and hold the bar together.

• Let f = f(x,t) represent these longitudinal forces, where the convention is adopted of choosing a positive value off to represent a force of compression, and a negative value to represent a force of tension.

• This choice of sign is the opposite of that conventionally taken by many material scientists but has the distinct advantage for us of making the compression of a solid by a positive increment of force analogous to the compression of a fluid by a positive increment in pressure.

• The stress in the bar of cross-sectional area S is defined as

Stress = f / SFor most materials, if the strain is small the

stress is proportional to it. This relationship is known as Hooke’s law,

f / S = -Y ( / x)𝜕𝜉 𝜕Where Y, the Young’s modulus or modulus

elasticity, is a characteristic property of the material.

(𝜕2 / x𝜉 𝜕 2)= 1/c2 ( 𝜕 2 / t𝜉 𝜕 2) Comparison of this equation with the corresponding for the transverse motion of a string that they have identical form, with longitudinal displacement 𝜉replacing transverse displacement y.

c2 = Y/p

The general solution has the same form as that for the transverse wave equation

𝜉(x,t) = 𝜉1(ct – x) + 𝜉2(ct + x)

The complex harmonic solution is

Note : 1. Since Young's modulus Y is measured under conditions allowing the strained rod to alter its transverse dimensions, (3.3.8) gives the phase speed only when the solid is a thin bar. 2. When the transverse dimensions of the solid are large compared to a wavelength, a combination of the bulk modulus and the shear modulus must be used in place of the Young's modulus to calculate the phase

• We are going to develop a simple linear wave equation for sound propagation in fluids (1D).

• In reality the acoustic wave equation is nonlinear and therefore more complicated than what we will look at in this chapter.

• However, in most common applications, the linear approximation to the wave equation is a good model. Only when sound waves have high enough amplitudes do nonlinear effects show themselves.

• What is an acoustic wave? It is essentially a pressure change. A local pressure change causes immediate fluid to compress which in turn causes additional pressure changes. This leads to the propagation of an acoustic wave.