4-5

1. The fluid moves, causing a change in density.

2. The change in density results in a change in pressure.

3. Differences in pressure leads to fluid motion.

The disturbances to the fluid are small relative to their equilib rium values, i.e., the acoustic pressure p ≪ P0 and the condensa tion s ≪ 1.

The system is linear, so the dependent variable only has first order terms and the superposition principle applies.

The equilibrium values are weak functions of time, i.e., although we know they are not constant over all time.

The fluid is inviscid, i.e., it has no viscosity.

The fluid is irrotational.

The pressure can be linked to the density and temperature of the f luid through what is known as the equation of state.

The continuity equation deals with the conservation of mass.

The Euler equation describes the relationship between a change in pressure and the velocity of the fluid.

The equations for acoustic wave propagation become linear under the assumption that the variations in pressure, density, and velocity caused by the wave are small compared to their equilibrium values. This is known as the linear acoustics approximation, and it simplifies the otherwise complex behavior of fluid motion.

The wave equation can also be formulated in the frequency domain, in which case it is referred to as the Helmholtz equation. It can be found by taking the Fourier transform of the wave equation.

Our assumption that the fluid is irrotational means that the vorticity of the fluid is zero, and thus the velocity field is conservative. There fore, we can define a scalar velocity potential field ψ and then calcu late the velocity field from this.

Yes, the wave equation assumes a conservative field as its basis.

• For acoustic waves, the pressure and density variations are governed by conservation principles such as:

  1. Conservation of Mass (Continuity Equation)

  2. Conservation of Momentum (Euler’s Equation)

  3. Conservation of Energy (Thermodynamic relations)

Fundamental forces like gravity and the electric force are conservative, and the quintessential example of a non-conservative force is friction. This has an interesting consequence based on our discussion above: If a force is conservative, it must be the gradient of some function.

The standard lossless, linear wave equation assumes idealized conditions. Modifying these assumptions introduces more complex wave equations:

1. Viscous Wave Equation: Includes elasticity and viscosity effects.

2. Refraction: The speed of sound varies spatially c=c(x), altering wave paths.

3. Dispersion: The speed of sound varies with frequency c=c(f), causing frequency-dependent distortion.

4. Non-linearity: The speed depends on pressure c=c(p), introducing complex wave interactions.

5. Attenuation: The speed becomes complex (c=c′+jc′′), where the real part represents wave propagation and the imaginary part models damping.

For most sonar applications, the lossless, linear wave equation suffices, but advanced models address these deviations for specific scenarios.

deep sound channel effect

1. Sound Speed Profile:

   • In the ocean, sound speed varies with temperature, salinity, and pressure.

   • Near the surface, warmer water increases sound speed, but deeper, colder water decreases it. Below a certain depth, increasing pressure raises sound speed again.

2. SOFAR Channel:

   • The SOFAR channel occurs where the sound speed reaches a minimum, typically at a depth of about 600–1200 meters.

   • Sound waves become trapped in this layer, bouncing between regions of higher sound speed above and below the channel.

   • This minimizes energy loss and allows sound to travel extremely long distances without significant attenuation.

3. Impact on Sonar:

   • Sonar signals transmitted in or near the SOFAR channel can propagate over vast distances with minimal loss, making it highly effective for underwater communication, navigation, and detection.

There are infinitely many possible solutions to the wave equation. To find a unique solution, we need to consider the domain or vol ume that the solution is defined on.

1. Dirichlet Condition: Specifies the value of the wave field at the boundary.

2. Neumann Condition: Specifies the gradient (normal derivative) of the wave field.

3. Robin Condition: A linear combination of Dirichlet and Neumann conditions.

4. Cauchy Condition: Combines Dirichlet and Neumann conditions.

5. Mixed Condition: Different conditions on different parts of the boundary.

Examples

• The sea surface is modeled as a Dirichlet boundary condition, where the pressure of the wave at the boundary is set to zero: p(x)=0∀x∈Sp(x) = 0

• The seafloor or a solid object like a submarine hull is modeled as a Neumann boundary condition, where the normal derivative of the wave field (particle velocity) is set to zero.

  • Context: At a rigid boundary, there is no particle motion perpendicular to the surface, which corresponds to a zero gradient of the wave field in the normal direction.

In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

1. Represent the source using f(x,t)

2. Recognize direct solutions are difficult.

3. Use Green’s function to solve for an impulse.

4. Combine impulse solutions to find the overall response.

5. Verify the solution satisfies the wave equation.

Green’s functions simplify solving complex wave equations by breaking them into manageable impulse responses.

It represents the response of the wave equation to an impulse source in free space, without boundaries or obstacles.