Lecture physics a2 interference phd pham tan thi

Superposing Sinusoidal Waves Superposing sine waves If you added the two sinusoidal waves shown, what would the result look like?. If we added the two sinusoidal waves shown, what would

Ho Chi Minh University of Technology

A Single Oscillating Wave

y(x, t) = Acos(kx !t)

The formula

direction

For a wave on a string, each point on the wave oscillates in the x

= 2⇡

! k

I / A2

The wavelength The speed/velocity

The intensity is proportional to the square of the amplitude

Multiple Waves: Superposition

individual wave.

For inequal intensities, the maximum and minimum intensities are:

I max = |A 1 + A 2 | 2

I min = |A 1 - A 2 | 2

Multiple Waves: Superposition

Constructive “Superposition” Destructive “Superposition”

Superposing Sinusoidal Waves

Superposing sine waves

If you added the two sinusoidal waves shown, what would the result look like?

- 1 0 0

- 0 5 0

0 0 0

0 5 0

1 0 0

- 2 0 0

- 1 5 0

- 1 0 0

- 0 5 0

0 0 0

0 5 0

1 0 0

1 5 0

2 0 0

100 200 300 400 500 600 700 800 900 1000

The sum of two sines having the same frequency is another sine with the same frequency.

Its amplitude depends on their relative phases.

Let’s see how this works.

If we added the two sinusoidal waves shown, what would the result look like?

Superposing Sinusoidal Waves

Superposing sine waves

If you added the two sinusoidal waves shown, what would the result look like?

- 1 0 0

- 0 5 0

0 0 0

0 5 0

1 0 0

- 2 0 0

- 1 5 0

- 1 0 0

- 0 5 0

0 0 0

0 5 0

1 0 0

1 5 0

2 0 0

100 200 300 400 500 600 700 800 900 1000

The sum of two sines having the same frequency is another sine with the same frequency.

Its amplitude depends on their relative phases.

Let’s see how this works.

If we added the two sinusoidal waves shown, what would the result look like?

Superposing sine waves

If you added the two sinusoidal waves shown, what would the result look like?

- 1 0 0

- 0 5 0

0 0 0

0 5 0

1 0 0

- 2 0 0

- 1 5 0

- 1 0 0

- 0 5 0

0 0 0

0 5 0

1 0 0

1 5 0

2 0 0

100 200 300 400 500 600 700 800 900 1000

The sum of two sines having the same frequency is another sine with the same frequency.

Its amplitude depends on their relative phases.

Let’s see how this works.

The sum of two sines having the same frequency is another sine with the same frequency → Its amplitude depends on their relative phases

Adding Sine Waves with Different Phases

y1 = A1cos(kx !t) and y2 = A1cos(kx !t + )

Spatial dependence

of 2 waves at t = 0

Resulting wave:

y = y1 + y2

A1(cos↵ + cos ) = 2A1cos

✓

↵ 2

◆ ✓

+ ↵ 2

◆

y1 + y2

( /2)

y = 2A1cos( /2)cos(kx !t + /2)

y = 2A1cos( /2)cos(kx !t + /2)

Amplitude Oscillation

Interference of Waves

What happens when two waves are present at the same place?

For equal A and ω:

A = 2A1cos(𝜙/2) I = 4I1 cos 2 (𝜙/2)

Lecture 2, p.6

Interference of Waves

What happens when two waves are present at the same place?

Always add amplitudes (pressures or electric fields).

However, we observe intensity (power).

For equal A and ω:

Stereo speakers: Listener:

2

A = 2A cos( / 2) φ ⇒ I = 4I cos ( / 2) φ

Constructive interference:

waves are “in phase”

(φ = 0, 2π, 4π, )

Destructive interference:

waves are “out of phase”

(φ = π, 3π, 5π, …)

Of course, φ can take on an infinite number

of values We won’t use terms like “mostly constructive” or “slightly destructive”

Stereo speakers:

Listener:

Terminology:

Constructive interference:

waves are “in phase”

(𝜙 = 0, 2π, 4π,…)

Destructive interference:

waves are “out of phase”

(𝜙 = π, 3π, 5π,…)

listener:

Lecture 2, p.7

Example: Changing phase of the Source

Each speaker alone produces an intensity of I1 = 1 W/m 2 at the listener:

Drive the speakers in phase What is the intensity I at the listener?

Now shift phase of one speaker by 90 o What is the intensity I at the listener?

φφφφ

I = I1 = A12 = 1 W/m 2

I =

I =

Drive the speakers in phase What is the intensity I at the listener?

Lecture 2, p.7

Example: Changing phase of the Source Each speaker alone produces an intensity of I1= 1 W/m 2 at the listener:

Drive the speakers in phase What is the intensity I at the listener?

Now shift phase of one speaker by 90 o What is the intensity I at the listener?

φφφφ

I = I1= A12 = 1 W/m 2

I =

I =

Now shift phase of one speaker by 90° What is the intensity I at the listener?

Lecture 2, p.7

Example: Changing phase of the Source

Each speaker alone produces an intensity of I1= 1 W/m 2 at the listener:

Drive the speakers in phase What is the intensity I at the listener?

Now shift phase of one speaker by 90 o What is the intensity I at the listener?

φφφφ

I = I1= A12 = 1 W/m 2

I =

I =

I = I1 = A12 = 1 W/m2

I =?

I =?

listener:

Lecture 2, p.7

Example: Changing phase of the Source

Each speaker alone produces an intensity of I1 = 1 W/m 2 at the listener:

Drive the speakers in phase What is the intensity I at the listener?

Now shift phase of one speaker by 90 o What is the intensity I at the listener?

φφφφ

I = I1 = A12 = 1 W/m 2

I =

I =

Drive the speakers in phase What is the intensity I at the listener?

Lecture 2, p.7

Example: Changing phase of the Source Each speaker alone produces an intensity of I1= 1 W/m 2 at the listener:

Drive the speakers in phase What is the intensity I at the listener?

Now shift phase of one speaker by 90 o What is the intensity I at the listener?

φφφφ

I = I1= A12 = 1 W/m 2

I =

I =

Now shift phase of one speaker by 90° What is the intensity I at the listener?

Lecture 2, p.7

Example: Changing phase of the Source

Each speaker alone produces an intensity of I1= 1 W/m 2 at the listener:

Drive the speakers in phase What is the intensity I at the listener?

Now shift phase of one speaker by 90 o What is the intensity I at the listener?

φφφφ

I = I1= A12 = 1 W/m 2

I =

I =

I = I1 = A12 = 1 W/m2

I = (2A1)2 = 4I1 = 4 W/m2

I =?