Since sound is generated by vibrations and travels in waves it can be modeled by trigonometric functions. When an object vibrates twice as fast, it produces a sound with a pitch one octave higher than before. Amplitude is associated with loudness. Suppose a sound can be modeled with the function g(è)= 15sin(200è)

a) Write a function for a sound with the same pitch as the sound modeled by g but 3 times as loud.

b) Write a function for sound with the same volume as the sound modeled by g but one octave higher.

c) Write a function for a sound with the same volume as the sound modeled by g but 2 octaves higher

Thanks for the help (this is due by May 13, 2011)

I will make it easy for you, nice me.

g(t)= Amplitude*sin (2PI*frequency*time)

That is the base form. In your case, amplitude is initially 15 and frequency is 200/2PI

What do you mean by

^three times as loud" ? If loudness is a measure of power per area, then the amplitude has to be higher by the square root of three.

To the ear, this will correspond to about 4.8 decibels more of sound level, not "three times as much".

To answer these questions, let's break down each part step by step:

a) To create a sound that is three times as loud as the sound modeled by g, we need to increase the amplitude of the original sound function.

The original function is g(x) = 15sin(200x).

To make the sound three times as loud, we need to multiply the amplitude, which is 15, by 3. Therefore, the new function will be:

f(x) = 3 * 15sin(200x).

Simplifying the equation gives:

f(x) = 45sin(200x).

So the function for a sound with the same pitch as the sound modeled by g but three times as loud is f(x) = 45sin(200x).

b) To create a sound with the same volume as the sound modeled by g but one octave higher, we need to increase the frequency of the original sound function by a factor of 2.

The original function is g(x) = 15sin(200x).

To raise the pitch by one octave, we need to double the frequency, which is 200. Doubling the frequency gives us 400.

So the new function will be:

h(x) = 15sin(400x).

Therefore, the function for a sound with the same volume as the sound modeled by g but one octave higher is h(x) = 15sin(400x).

c) To create a sound with the same volume as the sound modeled by g but two octaves higher, we need to increase the frequency of the original sound function by a factor of 4 (doubling it twice).

The original function is g(x) = 15sin(200x).

Doubling the frequency gives us 400. Doubling it again gives us 800.

So the new function will be:

k(x) = 15sin(800x).

Therefore, the function for a sound with the same volume as the sound modeled by g but two octaves higher is k(x) = 15sin(800x).

That's it! You now have the functions for sounds with different loudness and pitch based on the given model function.