1D Wave Equation
The 1D wave equation describes the behavior of waves propagating in one dimension, such as along a string or a rod. It is a partial differential equation that relates the second derivative of the wave function with respect to both time and position. The equation is fundamental in understanding the dynamics of wave motion and is widely used in various fields of physics and engineering.
5 Key excerpts on "1D Wave Equation"
eBook - ePubDesigning Quiet Structures
A Sound Power Minimization Approach
- Gary H. Koopmann, John B. Fahnline(Authors)
- 1997(Publication Date)
- Academic Press(Publisher)
...The equation of state for a liquid medium is identical to that given in Equation (1.19), but is derived in a slightly different manner, as discussed by Pierce (1989, pp. 30–34). The wave equation Equations (1.14), (1.15), and (1.19) represent a system of three linear partial differential equations in three dependent variables, p′, ρ′, and v′. The physical quantity which is most easily measured for a sound wave is the pressure, and thus the equations will be reduced to a single partial differential equation for the acoustic pressure. We take the partial derivative with respect to time of the linearized equation of conservation of mass [ Equation (1.14) ] giving (1.20) and take ∇ · of the linearized equation of the conservation of momentum [ Equation (1.15) ] giving (1.21) Subtracting Equation (1.21) from Equation (1.20) yields (1.22) or, after substituting for ρ′ from the linearized equation of state [ Equation (1.19) ], (1.23) To simplify the notation, we drop the superscript so that ρ is the small fluctuating component of the pressure, and v is the acoustic velocity. Also, we take the ambient density of the fluid to be ρ, without the subscript 0. Equation (1.23) can then be written as (1.24) which is the desired result for the wave equation. As a simple example, consider an acoustic wave which does not depend on the y or z coordinate. The wave equation then reduces to (1.25) which has the general solution (1.26) In Equation (1.26), F and G are arbitrary functions representing plane waves traveling in the positive and negative x–directions, respectively. 1.2 The Helmholtz Equation for Time–Harmonic Vibrations In many of the design methods that we discuss in this book, the surface vibration and associated sound field occur at a constant frequency, a typical example of which is the hum of a transformer at 120 Hz. After an initial transient period, the pressure field generated by the vibration will reach steady state and fluctuate at the same frequency as the excitation...
eBook - ePub- H. W. Liepmann, A. Roshko(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
...3.11 b may be approximated by a 1 2 (∂ / ∂x). For small disturbances, then, the exact equations of motion may be approximated by a set of equations that contains no non-linear terms, These are called the acoustic equations by virtue of the fact that the disturbances due to a sound wave are, by definition, very small. The corresponding approximations in the isentropic relations (Eqs. 3.10 b and 3.10 c) for a perfect gas give Either of the dependent variables may be eliminated from Eqs. 3.12. Since, cross differentiation gives and similarly This equation, which governs both and u, is called the wave equation. It is typical for phenomena in which a “disturbance” is propagated with a definite signal velocity or wave velocity. The signal velocity here, as we shall see, is a 1. The solution of the wave equation may be written in very general form. It is where F and G are arbitrary functions of their arguments. This may be checked by direct substitution in the wave equation: Let = x – a 1 t and η = x + a 1 t. Then where the prime denotes differentiation with respect to the argument. Continuing in this way, it may be shown that ∂ 2 / ∂t 2 = a 1 2 (F ″ + G ″) and ∂ 2 / ∂x 2 = F ″ + G ″, so that Eq. 3.14 a is satisfied. Similarly the solution for u may be written in terms of two arbitrary functions, Of course, f and g are related to F and G, for u must be related to by the original equations (3.12). These are satisfied if as may again be checked by direct substitution. 3.5 Propagation of Acoustic Waves The character of the solution (Eq. 3.15 a) may be illustrated by first taking G = 0, so that the density distribution at time t is given by This represents a disturbance, or wave, which at time t = 0 had the (arbitrary) shape and which now, at time t, has exactly the same shape, but with corresponding points displaced a distance a 1 t to the right (Fig. 3.2)...
eBook - ePub- Thomas L. Szabo(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
...In general, however, the shape of a wave will change in a more complicated way than these simple idealized shapes, which is why Fourier synthesis is needed to describe a journey of a wave. Figure 3.1 Plane, cylindrical, and spherical waves showing surfaces of constant phase. In order to describe these basic wave surfaces, some mathematics is necessary. The next section presents the essential wave equations for basic waves propagating in an unbounded fluid medium. In order to characterize simple echoes, following sections will introduce equations and powerful matrix methods for describing waves hitting and reflecting from boundaries. 3.2.2 Wave Equations for Fluids In keeping with the common application of a fluid model for the propagation of ultrasound waves, note that fluid waves are of a longitudinal type. A longitudinal wave creates a sinusoidal back-and-forth motion of particles as it travels along in its direction of propagation. The particles are displaced from their original equilibrium state by a distance or displacement amplitude (u) and at a rate or particle velocity (v) as the wave disturbance passes through the medium. This change also corresponds to a local pressure disturbance (p). The positive half cycles are called “compressional,” and the negative ones, “rarefactional.” If the direction of this disturbance or wave is along the z axis, the time required to travel from one position to another is determined by the longitudinal speed of sound c L, or t = z/c L...
eBook - ePub- David M. Scott(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...A simple pulse is shown in Figure 3.5c, and an asymmetric pulse in Figure 3.5d. FIGURE 3.5 Four varieties of wave shape, as discussed in the text. In suspensions (solid particles suspended in liquid) and solid mixtures (e.g., concrete), the longitudinal wave is dissipated (attenuated) when it scatters from the particles; measurements based on this effect are discussed in chapter 8. Even in the absence of particles to scatter the sound, both solids and fluids have an intrinsic attenuation that reduces the sound intensity over distance. 3.3 The Wave Equation and Its Solutions It can be shown (see section 3.6) that a sound wave ψ(x,t) is a solution of the wave equation: ∂ 2 ψ ∂ x 2 = 1 c 2 (∂ 2 ψ ∂ t 2) (3.2) where x is position, t is time, and c is the speed of sound. Solutions to equation 3.2 are periodic functions and linear combinations of periodic functions. The sine function is one of the most familiar solutions to the wave equation; the displacement at a fixed position for the wave in Figure 3.4a can be represented by the function Asin(ω t), and the wave in Figure 3.4b can be described by Asin(kx). In general a continuous wave ψ, which is a function of both position and time, can be represented by either of these two equations: ψ (x, t) = A sin (k x − ω t + φ) (3.3a) ψ (x, t) = A sin (k x + ω t + φ) (3.3b) where (at least for now) A, k, ω, and φ are constants. It is easily demonstrated by substitution that equations 3.3a and 3.3b are solutions of equation 3.2 if and only if c = ω k (3.4) The quantity c is the velocity of the wave, which is a material property of the medium in which the wave travels. Equation 3.3a represents a wave traveling to the right, and equation 3.3b represents a wave traveling to the left. The constants in equation 3.3 determine the size, wavelength, speed, and phase of the wave. The amplitude of the wave is given by A, so the peak-to-peak amplitude equals 2A as shown in Figure 3.4a...
eBook - ePub- Thomas L. Szabo(Author)
- 2004(Publication Date)
- Academic Press(Publisher)
...The practice of applying a fluid model to tissues involves using tabular measured values of acoustic longitudinal wave characteristics in the previous equations. The main difference between waves in fluids and solids is that only longitudinal waves exist in fluids; many other types of waves are possible in solids, such as shear waves. These waves can be understood through electrical analogies. The main analogs are stress for voltage and particle velocity for current. The relationships between acoustic variables and similar electrical terms are summarized in Table 3.1. Correspondence between electrical variables for a transmission line and those for sound waves along one dimension in both fluids and solids enables the borrowing of electrical models for the solution of acoustics problems, as is explained in the rest of this chapter. Note that for solids, stress replaces pressure, but otherwise all the basic relationships of Eqs. (3.1 – 3.4) carry over. Another major difference for elastic waves in solids in Table 3.1 is the inclusion of shear waves. Waves in solids will be covered in more detail in Section 3.3. TABLE 3.1 Similar Wave Terminology 3.2.3 One-Dimensional Wave Hitting a Boundary An important solution to the wave equation can be constructed from exponentials like those of Eq. 3.9. Consider the problem of a single-frequency acoustic plane wave propagating in an ideal fluid medium with the characteristics k 1 and Z 1 and bouncing off a boundary of different impedance (Z 2), as shown in Figure 3.2. Assume a solution of the form, Figure 3.2 One-dimensional model of wave propagation at a boundary. (3.19) which satisfies the previous wave equation. RF is a reflection factor for the amplitude of the negative-going wave. An electrical transmission line analog for this problem, described in more detail shortly, is symbolized by the right-hand side of Figure 3.2. The transmission line has a characteristic impedance (Z 1), a wavenumber (k 1), and a length (d)...
