Deriving The Wave Equation - So far in this book we have dealt with vibrations. These are
processes that vary as functions of time.We were able to describe relevant
types of vibrations with common differential equations.
This chapter now focuses on waves, which are processes that vary with both time and space. Their mathematical description requires partial differential equations. Motivated by the electroacoustic analogies, we start this chapter with an excursion into electromagnetic waves. This excursion will bring us back to sound waves in the next section1. Deriving The Wave Equation
This chapter now focuses on waves, which are processes that vary with both time and space. Their mathematical description requires partial differential equations. Motivated by the electroacoustic analogies, we start this chapter with an excursion into electromagnetic waves. This excursion will bring us back to sound waves in the next section1. Deriving The Wave Equation
Equivalent circuit for a differential section of a homogeneous,
lossless electrical transmission line
Figure 7.1 illustrates the equivalent circuit for a elementary
section of a homogeneous,
lossless electrical transmission line. For this section, the
following loop and node equations hold, 7 The Wave Equation in Fluids
The following two linear differential equations are obtained by
neglecting higher-order differentials, This
set of equations shows that a temporal variation of the slope of one of the variables results in a proportional spatial variation of the
slope of the other. This causes that the total energy on the line swings
between two types of complementary
energies, namely,
Deriving The Wave Equation
• magnetic energy per length, W_ = 1 2L_ i2
• electric energy per length, W_ = 1 2C_ u2
The two linear differential equations (7.3) can be combined into a
differential equation of second order, which is
This is the so-called wave equation, here in the formulation
for electromagnetic waves on an
electrical transmission line. Hereby c line, el is the propagation speed of
electromagnetic waves on the transmission line, namely,
In acoustics we also have waves that propagate along one coordinate,
for example, in gas-filled tubes with small diameters compared to wavelength.Such
longitudinal compression waves are schematically sketched in Fig
Deriving The Wave Equation, One-dimensional longitudinal acoustic wave. Zones of compression
and rarefaction are schematically indicated Two complementary forms of energies are as well required for these
types of waves to occur, but in this case we choose energies per volume to be
compatible with the use of p and v as characteristic sound-field
quantities. We thus get 7.1 Derivation of the One-Dimensional Wave Equation 89
- potential energy per volume, W__ = 1 2κp2 κ ...volume compressibility
- kinetic energy per volume, W__ = 1 2_ v2 _ ...mass density
Mimicking the wave equation for the electromagnetic waves, we now suggest
the following wave equations for one-dimensional acoustic waves,
and the
combined second order expression as
with the
propagation speed of sound waves to be
The pressing question now is whether these supposed equations, which
are so neatly analogous to electric
expressions, are actually in compliance with physical reality.
The answer is yes, at least approximately. Yet, there are a number of features of the media where sound exists that must be idealized. The following section will elaborate on them. Deriving The Wave Equation
The answer is yes, at least approximately. Yet, there are a number of features of the media where sound exists that must be idealized. The following section will elaborate on them. Deriving The Wave Equation
Sourch : Acoustic Engineering Book
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