Standing WavesBy Rick Grootes

We will be viewing all of these problems and equations with a phase constant (ϕ0) of zero for both waves.

Remember, this could not be the case for many examples, as not all waves are as perfect as we would hope for them to be.

KEY POINT

They are two harmonic waves of equal amplitude, wavelength, and frequency moving in the opposite direction of one another

Mathematically, we see these two waves as:

D1(x,t)=Asin(kx-ωt)

and:

D2(x,t)=Asin(kx+ωt)

where:

D(x,t)=D1(x,t)+D2(x,t)

=A[sin(kx-ωt)+sin(kx+ωt)] The sum of these two waves gives rise to a simple trigonometric identity similar to:

sin(a-b)+sin(a+b)=2sin(a)cos(b)

In this case: D(x,t)=2Asin(kx)cos(ωt)

What are standing waves?

Using the formula we found in the previous slide we can define a position-dependent amplitude A(x):

A(x)=2Asin(kx)=2Asin(2π*x/λ)

we already know from our formula sheet that:k=(2π/λ)

Using the above definition, we can change around our formula to:

D(x,t)=A(x)cos(ωt)

Taking it apart!D(x,t)=2Asin(kx)cos(ωt)

Please find the amplitude and wavelength of the following examples:

1) A(x)=(80.0cm)sin(8.00x)

2) A(x)=(35.0cm)sin(1.57079x)

3) A(x)=(5624.0cm)sin(0.0981748x)

Please find answers on next slide, but don’t skip!

Lets do a little tester!

1) 2A=80.0cm A = 40.0cm2π/λ=8.00 λ = π/4 = 0.7854 m

2) 2A=35.0cm A = 17.5cm

2π/λ=1.57079 λ = 4.00 m

3)2A=5624.0cm A = 2812.0 cm

2π/λ=0.0981748 λ = 64.0 m

Answers to Slide 5

These waves are different because the x (position) is part of the sine function, whereas the t (time) is part of the cosine function.

They have nodes and anti-nodes: Nodes: certain points on the string that have zero

amplitude and remain at rest at all times Anti-nodes: points that move with the maximum

possible amplitude of 2A.

How are these waves different?

At the nodes of a standing wave, A(x)=0.

Therefore, they occur when:

sin(2π*x/λ)=0

(2π*x/λ)=mπ where m=±1,±2,…

x=m(λ/2) where m=±1,±2,…

x=0,±(λ/2),±(λ),±(3λ/2),±(2λ),…

How do we find the location of nodes and anti-nodes:

We already know that anti-nodes occur when A(x)=±2A. This happens when:

sin(2π*x/λ)=±1

(2π*x/λ)=(m+½)*π where m=0,±1,±2,…

x=(m+½)*(λ/2) where m=0,±1,±2,…

x=±(λ/4), ±(3λ/4), ±(5λ/4),…

Now for Anti-nodes:

When we think about it, two consecutive anti-nodes will always be half a wavelength apart.

Similarly, an adjacent node and anti-node will always be a quarter wavelength apart.

For simplicity

We all know from our trusty formula sheet that: Period(T) and frequency(f)

where T=(1/f)

and that angular frequency (ω):

ω=(2π/T)=(2π*f)

Therefore, in full, our equation looks something like:

D(x,t)=2Asin(2π*x/λ)cos(2π*t/T)

We can’t forget about cosine

John and Cindy took a trip to Scotland and wanted to see some of the greatest cliffs in the world. As they peered over the edge, they saw huge waves crashing against the cliff and bouncing off to give even larger waves to give what seemed to look like standing waves. With their knowledge of physics, they created an equation for these huge waves and thankfully, recorded it all on their handy-dandy water-proof paper! Go to the next slide to see all their findings.

Lets take a trip to the rocky coast of Scotland(Info Part 1)

Part 1)

They found that the amplitude of the waves before they formed a standing wave were 20.0m high a piece.

a) What was the amplitude of the standing wave?

They also found that once the waves merged, they had nodes at the cliffs (x=0), 15.0m away from the cliffs and 30.0m away from the cliffs.

b) What was the wavelength of the standing wave?

Collected Data Part 1

Part 2)

Then, they wanted to rearrange the data they collected into the equation:

A(x)=2Asin(kx)

Data Collected Part 2

While they took note of the amplitude and enjoyed the sights, they also pulled out their stop watches and calculated how many anti-nodes they saw in 8 seconds. In those 8 seconds, the saw exactly 24 anti-nodes.

a) what is the frequency of the waves?

b) what is the period of the waves?

c)use all of the information gathered to formulate an answer to the equation:

D(x,t)=2Asin(kx)cos(ωt)

(Info for Part 3)

See next page for solutions to Part1, Part 2 and Part 3!

THE END