WavesTopic 4.5

Wave Properties

Wave Behaviour

Reflection in one dimension

This diagram shows a pulse travelling along a string

This diagram shows the pulse after it has been reflected

Notice

The pulse keeps its shape It is inverted It has undergone a 180o phase

change, or change in phase

Fixed vs Free-end Reflection

The inversion is because the instant the pulse hits the fixed end, the rope attempts to move the fixed end upwards

It exerts an upwards force on the fixed end By Newton’s third law, the fixed end will exert

an equal but opposite force on the rope This means that a disturbance will be created

in the rope which is “downwards” and will start moving to the left

If the end of the rope is not fixed but free to move the situation is different

Most of the pulse would carry on in the same direction, some would be reflected but the reflected pulse is in the same phase as the original pulse

There is a change of direction, but no inversion here

Fixed vs Free-end Reflection

Wave Behaviour

Reflection in two dimensions

Reflection

Wavefronts incident upon a boundary…

Incident wavesReflected waves

Normal

Angle ofincidence

Angle ofreflection=

The Law for Reflection

• The angle of incidence is equal to the angle of reflection

-keep in mind that angles are measured with respect to the normal at point of contact

• Also - The incident ray, the reflected ray and the normal lie on the same plane

• Use this rule for any ray or wave diagram involving reflection from any surface

• For circular waves hitting a flat reflector, the reflected waves appear to come from a source, which is the same distance behind the reflector as the real source is in front of it

• Also a line joining these 2 sources is perpendicular to the reflecting surface

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• If a plane wave is incident on a circular reflector then the waves are reflected so that they

–Converge on a focus if the surface is concave

–Appear to come from a focus if the surface is convex

Incident wavefronts

Reflected wavefronts(semi circles)

Incident wavefronts

Reflected wavefronts

Wave Behaviour

Refraction

• The speed of a wave depends only on the nature and properties of the medium through which it travels.

• Refraction is the change of direction of travel of a wave resulting from a change in speed of the wave when it enters the other medium at an angle other than right angles.

• In a ripple tank this is achieved by using a flat piece of plastic, giving two regions of different depth

• As the wave passes over the plastic it enters shallow water and slows down.

• Remember: v = f • If v decreases and f is constant (the source

hasn’t changed) must also decrease, so the waves get closer together

• If the waves enter the shallow area at an angle then a change in direction occurs.

Shallow water

• This is because the bottom of the wavefront as drawn, hits the shallow water first so it slows, and hence travels less distance in the same time as the rest of the wavefront at the faster speed travel a larger distance!

Deep water

• If the waves enter the deep area at an angle then a change in direction occurs

• This is because the top of the wavefront hits the deep water first so it speeds up, and hence travels more distance in the same time as the rest of the wavefront at the slower speed travel a smaller distance!

Refraction for light

Partial reflection

Incident ray

Incident rayRefracted ray

Refracted ray

Partial reflection

Snell’s Law

• Snell discovered that for any two media

•Sin 1 / Sin 2 = constant

• The constant is…the ratio of the wave velocity in the two media v1 / v2

• Where 1 is the angle of incidence in the 1st medium, v1 is the velocity in that medium

• And 2 is the angle of refraction in the second medium, v2 is the velocity

• Therefore

• Or… n1 sin 1 = n2 sin 2

• Where n is the index of refraction of the media

• For light and optics…

nvacuum = 1 (also taken as the value in air)

• This law enable us to define a property of a given optical medium by measuring 1 and 2 when medium 1 is a vacuum

• The constant is then the property of medium 2 alone and it is called the refractive index (n).

• We usually write • n = (Sin i) / (Sin r)

• n is also a ratio of the speeds in the 2 mediums i.e. n = cvacuum / vmedium

Diffraction

• Diffraction is the spreading out of a wave as it passes an obstacle or through an aperture (an opening)

• When the wavelength is small compared to the aperture the amount of diffraction is minimal. When the wavelength is comparable to the size of the opening then diffraction effects are significant.

• Diffraction also takes place when a wave moves passed an obstacle

• If the wavelength is much smaller than the obstacle, little diffraction takes place

• If the wavelength is comparable to the obstacle size, then diffraction is significant

Huygens’ Principle

• Christian Huygens' idea was to consider every single point on the wavefront of the wave as the source of a new wave disturbance.

• In other words a point on the wavefront would emit a spherical “wavelet” or secondary wave, of same velocity and wavelength as the original wave.

• Therefore as a wave goes through a gap or passed an obstacle the wavelets at the edges spread out the wave energy.

• Huygens’ construction can be used to predict the shapes of the wave fronts.

Huygens’ Principle

• Huygens’ Principle

• The new wavefront would then be the surface that is tangent to all the forward wavelets from each point on the old wavefront.

• We can easily see that a plane wavefront moving undisturbed forward easily obeys this construction.

The Principle of Linear Superposition

• Pulses and waves (unlike particles) pass through each other unaffected and when they cross the total displacement of the medium is the vector sum of the individual displacements due to each pulse at that point.

Interference

• Most of the time in Physics we are dealing with pulses or waves with the same amplitude.

• If these cross in a certain way we will get full constructive interference, here the resultant wave is twice the amplitude of each of the other 2

+ =

• If the pulses are 180o () out of phase then the net resultant of the string will be zero. This is called complete destructive interference.

+ =