# Acoustic pressure wave from a spherical explosion

• markmai86
In summary, the conversation discusses a function representing the acoustic wave from a spherical explosion and its possible solution for the wave equation. It also explores the corresponding velocity field and suggests a modification to the pressure wave to eliminate residual velocity. The conversation also mentions the wave equation and its equations.
markmai86

## Homework Statement

The acousitc wave from a sperical explosion is sometimes represented by a function:

p = 0 for t < r/c
p = B / r * exp [ - (t-r/c) / t0 ] for t > r/c (or equal)

a) Does this represent a possible soltion for the wave equation?
b) Find the corresponding velocity field, in particular show that there is a finite residual velocity B*t0/(rho*r^2) when t goes to the infinite.
c) Draw the corresponding wave diagram
d) Suggest a modification for the pressure wave, such that the residual velocity is zero.

## Homework Equations

Wave equation:

d^2 psi / dt^2 = c0^2 * nabla^2 * psi

## The Attempt at a Solution

a) No, this does not represent a possible solution for the wave equation. The wave equation is a differential equation and the function given does not meet these requirements. b) The corresponding velocity field for this pressure wave is:v = B * t0 / (rho * r^2) when t goes to infinity. c) The corresponding wave diagram is a graph of the pressure wave against time. d) A modification for the pressure wave, so that the residual velocity is zero, would be to modify the function to include a damping factor at the end. This could be done by adding an exponential term at the end of the pressure wave equation.

a) Yes, this function represents a possible solution for the wave equation. It satisfies the wave equation for both t < r/c and t > r/c, and also satisfies the boundary conditions for a spherical explosion.

b) The corresponding velocity field can be found by taking the first derivative of the pressure function with respect to time:

v = - (B / r^2) * exp [ - (t-r/c) / t0 ] * (1/t0)

As t approaches infinity, the exponential term goes to zero, leaving a finite residual velocity of B*t0/(rho*r^2).

c) The corresponding wave diagram would show a spherical wave expanding outwards from the point of explosion, with the amplitude decreasing as it travels further away from the source.

d) To modify the pressure wave such that the residual velocity is zero, we can add a term to the pressure function that cancels out the exponential term as t approaches infinity. This can be achieved by adding a constant value to the pressure function, such as p = B / r * exp [ - (t-r/c) / t0 ] + A, where A is chosen such that the residual velocity is zero. This would result in a modified pressure wave with a flat velocity profile at infinity.

## 1. What is an acoustic pressure wave?

An acoustic pressure wave is a type of sound wave that is created by a sudden disturbance or explosion. It is a compression wave that travels through a medium, such as air or water, and causes temporary changes in pressure.

## 2. How is a spherical explosion different from other types of explosions?

A spherical explosion is characterized by an outward expansion of energy in all directions from a central point, resulting in a spherical shock wave. This is in contrast to other types of explosions, such as directional or cylindrical explosions, where the energy expands in a specific direction.

## 3. How does the size of a spherical explosion affect the acoustic pressure wave?

The size of a spherical explosion can greatly impact the intensity and duration of the acoustic pressure wave. A larger explosion will create a stronger and longer-lasting pressure wave, while a smaller explosion will produce a weaker and shorter-lived pressure wave.

## 4. What factors can affect the propagation of an acoustic pressure wave?

The propagation of an acoustic pressure wave can be influenced by several factors, including the strength and location of the explosion, the surrounding environment (e.g. air density, temperature), and obstacles in the wave's path (e.g. buildings, mountains).

## 5. Can acoustic pressure waves from a spherical explosion be harmful to humans?

Yes, depending on the size and proximity of the explosion, acoustic pressure waves can be harmful to humans. Exposure to high levels of acoustic pressure can cause damage to the eardrums and hearing loss, as well as other physical injuries such as lung damage. It is important to take necessary safety precautions when dealing with explosions and their acoustic pressure waves.

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