Part 2 - waves in 1D

Assignment 1 by Corinne Lee 45672003

This section is on waves in one dimension. The previous section discusses the concepts of shifting and scaling graphs. This is essential to this section where the fundamental wave equation is formed by these transformations. The image below is a simple cosine wave.

A wave is formed by simple periodic motion (in space and time). Simple waves have several components; wavelength (), velocity (c), time period (1/frequency=1/), amplitude (A) and frequency ().

• wavelength: spatial distance between two crests.
• velocity: is the speed at which the crests travel.
• time period: the time it takes to go through on cycle at a fixed point.
• amplitude: the height of the wave.
• frequency: the number of waves that crosses a certain point per second.

Wavelength

The wavelength() of a wave is the distance from one crest to the next.

Velocity

Velocity is denoted by the letter c. The green arrow in the diagram below represents the velocity (a vector) of the wave. It is given by the formula.

c=/T

where T=time
=wavelegth

The faster the wave the smaller the wavelength.

The slower the wave the longer the wavelength.

Time Period

The time period of a wave is the time it takes one wavelength to pass a point ie how long does it take for the green dot to pass the yellow line.

1/frequency=1/

Amplitude

The amplitude of the wave is the distance from the x-axis to the crest.

Frequency

The frequency() is the number of crests that cross a point in a given amount of time. ie how many crests pass the yellow line in time T.

There are two types of frequencies

1. period per second
2. radians per second

When we use radians per second and given , c and we get the formula.

T=2/

where T=time This changes the velocity formula to

c=/2

The wave in one dimension and the wave equation

This is where shifting and scaling plays a role in forming the wave function.  At time T=0 we have the simple cosine wave with the wavelength at 2. f(x)=cos(x) At time T the graph has shifted to the right by cT. f(x)=cos(x-cT) By horizontally scaling the graph we can change the wavelength. we want to scale x by /2 so using scaling we divide x by what we want to scale. f(x)=cos(x-cT)//2 =cos2/(x-cT) or f(x)=cos(2x/-cT2/) Finally by scaling vertically we can change the amplitude. We scale by A. f(x)=Acos(2x/-cT2/) This is the wave equation and by varying wavelength (), velocity (c), time period (1/frequency=1/), amplitude (A) and frequency () we can change the equation and change the wave.