## Doppler Effect

Imagine a Formula 1 car approaching the stands at 60m/s. The frequency of sound made by the engine as heard by a stationary observer in the stands is higher than the actual frequency as heard by the driver. The sound is squashed up – or better, the apparent wavelength is decreased and the apparent frequency increased.

As the car recedes away from the stands, exactly the reverse happens. the observer waves goodbye to the red line. Think of EEEEYOWWWW as the car approaches then recedes.

For a stationary observer and a moving source, we can write:

These will mostly do – but IB requires us to use these as well:-A quick calculation shows how the first equation works. Let the car be moving towards us in the stands at a speed us of 60m/s and emitting a frequency f of 800Hz.

Speed of sound in air is 340m/s. We can find the frequency f’ as heard by the stationary observer. Common sense tells us whether we add or subtract the velocities – in this case, we subtract and hear a higher frequency as it approaches us. (the EEEE bit)

As it recedes, we add, thus: (the YOWWW bit)

Police speed detectors bounce microwave radiation (about 10GHz) off a moving vehicle and detect the reflected waves. Because the car is moving towards the police observer, these waves are shifted in frequency by the Doppler effect and the difference in frequency between the transmitted and reflected waves provides a measure of the vehicle’s speed. Of course it works just as well for recession speeds as well.

Two Doppler shifts because of the reflection from a moving target. c is of course the speed of light

By observing distant galaxies, Edwin Hubble concluded that distance and recession speed were proportional – so galaxies further away are receding faster than closer galaxies. We know this because the atomic fingerprint or spectrum of atomic hydrogen or helium is shifted to the red (long wavelength) end of the visible spectrum. The degree of redshift can be used to find out how far away a galaxy is.

This absorption spectrum shot (idealised) shows what the spectrum of atomic hydrogen might look like from several distant objects like galaxies. The further away, the greater the redshift. Redshifts of up to 0.95c have been observed – the light having taken almost the lifetime of the Universe to reach us.

Finally, a medical use. Doppler blood flow is a technique whereby ultrasound waves (f about 800Hz) emitted from a piezoelectric transducer (transmitter/receiver) are reflected off red blood cells in an artery or vein as they are moving towards the stationary detector. The more occluded or blocked the artery is (think about a fluid in a pipe) the faster the cells are moving. It can also be used to find blood clots in deep veins – DVT – deep vein thrombosis – can be fatal.

The detector and the moving cells are at an angle hence the cosine term and, like the police car, the factor 2 accounts for the reflection from a moving source.

## Ultrasonic Techniques in Medicine

Exam questions about this are sometimes comparative. X rays are more invasive since they cause ionization. Ultrasound (US) is a non-invasive procedure causing minimal local heating which relies on the fact that sound waves well above the range of human hearing are reflected at a boundary, such as that between bone and air, for example. The greater the density difference between the two media, the higher the percentage reflected back and vice-versa.

The speed with which an acoustic wave moves through a medium is dependent upon the density and elastic resistance of the medium. Media that are dense will transmit a mechanical wave with greater speed than those that are less dense. As an example, the acoustic speed of a mechanical wave through air is about 340m/s, through water, it is 1500m/s, through soft tissue, 1540 m/s and through bone, 4080m/s, twelve times faster than in air. It used to be though that ultrasonic resolution was poorer than X-rays, this is now not in fact the case; foetal heartbeat and retinal scanning requires a resolution of the order of 1-2mm and this is achieved by raising the US frequency. For obstetric work, between 2 and 7 MHz is fine, for retinal work, up to 15MHz is used which increases resolution substantially. A rule of thumb is that the organ under study should be about 200 wavelengths away from the transducer (or emitter/receiver)

When an AC source of very high frequency is applied to a piezoelectric crystal (even quartz works fine) it vibrates, creating an ultrasound wave at the same frequency as the AC. The sound produced is then directed at an object and then bounces back off the object under investigation. When the sound wave comes back to the   piezoelectric crystal, it has the reverse effect – causing the mechanical energy produced from the sound vibrating the crystal to be converted into electrical energy. By measuring the time between when the sound was sent and received, the amplitude of the sound and the pitch of the sound, a computer can produce images, calculate depths and calculate speeds.

We recall having previously mentioned specific acoustic impedance, Z, the product of material density and velocity in the medium.

Now consider the interface between two materials of specific acoustic impedance Zand Z2, having different densities and therefore speeds, we may write:

This is worth a moment’s thought, since it suggests the most reflection, hence largest signal received, will happen when the sound is reflected off the interface between two materials of very different densities.

In obstetrics, if a coupling gel were not used to move the transducer across the patient’s abdomen, apart from the lack of lubrication, all the sound would be reflected and none transmitted into the body. This “acoustic coupling” is vital to ensure that most of the sound isn’t reflected back and so gets past the air/skin barrier.

A Scan

A scans can be used in order to measure distances. A transducer emits an ultrasonic pulse and the time taken for the pulse to bounce off an object and come back is graphed in order to determine how far away the object is. A-scans only give one-dimensional information and therefore are not useful for imaging.This is the simplest kind. A burst of ultrasound is passed into, for example, a newborn foetal skull. If the two halves of the brain are equally sized, the midline echo will occur exactly between the reflections on the opposite sides of the skull. If not, the echoes are skewed to one side (see diagram. T = transducer and we imagine looking down at the head from the top. Not great but you hopefully get the idea)

Retinal detachment can be detected using high frequency US, as shown.

In a healthy retina, the middle spike should be absent but because the retina (yellow line) is not attached to the eyeball surface and is floating in the fluid in front, an echo is seen as the US bounces off it.

B Scans

B scans are multiple A scans, produced by a moving transducer. They take many A images per second and an image intensifier – these days a computer – retains and displays the information in real time, building up a 2D slice across the transducer’s moving path. It is a routine procedure in obstetrics where the transducer is moved rapidly over the abdomen from right to left and back again a few times so as to collect enough data to build up an image. This image shows twin foetuses.

If the transducer can be placed towards an oncoming blood vessel, the changing velocity of the red blood cells, hence overall speed of blood can be monitored as the US signal bounces off oncoming red blood cells and the consequent Doppler shift measured. Notice the use of the cosine since the blood velocity along the detector line between source (S) and emitter (E) is

where u=blood velocity,

c=velocity of sound in blood

and f0 is the incident ultrasound frequency

You should study carefully the “Advantages and Disadvantages” Table I2.2 on p 709