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The last few weeks have been busy ones for me, and lately I find myself tapping the end of my patience reservoir. I was fortunate, however, that just at a time when I thought I was in danger of destroying inanimate objects, I found myself at a weekend-long bluegrass festival. Say what you will about the genre -- I will attest to the calming power of almost any live acoustic music if nothing else. As for bluegrass, it’s as Steve Martin said: “You can't play a sad song on the banjo, it always comes out so cheerful.” I found myself relaxed, my mood improved, and I spent some quality time with my dad.
One of the bands featured a musician who burned some tuning time by imitating commonly heard city and wilderness sounds on his fiddle. Along with bird calls, car-whooshings and train whistles, he included a pretty impressively precise rendition of an police siren passing by. Impressed, I nudged my dad and whispered, “That was a pretty accurate Doppler Effect.” He responded, “You should write about that in your post for next week.”
Who am I to deny my most loyal fan?
So now that I’ve wasted your time by making you read that irrelevant introduction, I give you a discussion on perhaps the single most recognized physical phenomenon – the one that helps us predict the weather, identify distant stars and galaxies, and medically image the flow of blood through the heart. Understanding it helps us understand supersonic speeds (and sonic booms that accompany them), and most importantly, the it is the scientific principle that tags us with expensive speeding tickets: The Doppler Effect.
The two kinds of waves
A wave can travel in in two different ways – longitudinally or transversally. To get a picture of these two types of waves, imagine that you have attached a slinky to a wall. If you were to shake the slinky upward, you would create a wave crest. This crest would travel along the length of the slinky, towards the wall and away from your hand. That means that the crest of the wave (up and down) is perpendicular to the direction that the wave is travelling (towards the wall). This is a transverse wave. The word transverse has several different definitions, but it usually refers to something that is oriented cross-wise, or something that comes from the side.
The second type – the longitudinal wave – similarly travels from your hand to the wall, though the “crests” exist in the same plane as the direction of travel. Imagine that instead of shaking the slinky up and down, you push it forward. A compression is created that travels away from your hand. These compressions and their corresponding refractions travel along the slinky, towards the wall. These waves are also known as compression waves, for obvious reasons. Sound is a compression wave, where slinky is replaced with air as the wave medium. A physical object (a tuning fork, for example) vibrates. This physical vibration pushes against the surrounding air molecules, which travel outward, bumping up against each other in a compressing and retracting pattern that has the same frequency as the object. When these air molecules reach us, they vibrate little hairs in our ears and we hear notes.
The Doppler Effect for sound
So you’re walking along the sidewalk, and some jerk comes driving by you just absolutely leaning on the horn. When you first hear that ear-splitting screech approaching you from down the street, the horn is high-pitched. As it gets closer, not only does it get louder, but the pitch seems to drop. It comes flying by you and in the instant that it passes by, you hear the true pitch of the horn. Then as it shoots off, moving away from you, the tone gets lower. It sounds like this:
This is the Doppler effect, which was described in 1842 by Austrian physicist Christian Doppler. The frequency that the observer hears is different than the emitted frequency, and the change is based on the velocity at which either the observer or the source is travelling. The equation is:
Where v is the normal speed of sound in air (340 meters per second or about 750 mph), vr is the velocity of the receiver, vs is the velocity of the source, fo is the original frequency of the sound being created, and f is the sound you hear. The velocity of the receiver is considered positive if you are moving towards the source (negative if you’re getting further away) and the source velocity is considered negative if its moving towards you (positive if its moving away). For those of you afraid of equations, what this means is that if you and the source are getting closer together, the frequency of sound that you hear will be higher than the frequency of sound emitted, which translates to higher pitch. If you and the source are getting farther apart at, then the frequency you hear will be lower than the sound emitted. Note that this applies for objects that travel at constant speeds.
A common misconception
Something that is often poorly explained is the gradual tone-slide in the sound that you hear. The change happens gradually as the object passes by, but astute readers (or those who were spared from the common affliction of math-trauma) might notice something odd about the above equation. We’re calculating the frequency of the sound we hear by multiplying the original frequency by a number that is calculated from a bunch of velocities. If the speed of sound in air is constant (which it approximately is), and the speed you’re traveling at is constant (if you’re standing still, it’s a nice healthy zero), and the speed that the vehicle is traveling at is constant (not accelerating or decelerating at all), then that number generated within the parentheses will be -- can you guess? -- constant. The frequency of the horn’s sound is always constant, so why do we hear the frequency change gradually as the car goes by? Shouldn’t it just be a constant?
The explanation for this is that the car isn’t traveling directly at you, it’s traveling by you. While its velocity in its direction of travel is always the same, the component of velocity that is aimed towards you is reaching you at an angle, and as that angle gets larger, the component of that points from the car to you gets smaller.
Mach speed and sonic booms
In order to explain this sub-topic, I’ll use the help of another brilliant animation that illustrates the Doppler effect discussed above. In these images, the center dot represents the source, and the circles that spread out from it represent the peaks of the sound waves as they propagate outward. The left-hand side illustrates a stationary source, while the right-hand side illustrates a source that is moving to the right. Notice that the wavefronts are closer together (a higher frequency) to the right of the source, while the wavefronts are more spread out (lower frequency) on the left of the source.
But what if the source was moving as fast as the sound waves could propagate outward – the speed of sound? The picture then looks something like this:The crests on the right are now all bunched up together along the path of the object, and no sound waves are travelling in front of it. All these bunched-together crests create an intense wall of high pressure (since remember, sound is a result of compressing air molecules). This pressure front is so intense that it’s more of a shockwave, and what you hear as it passes is not a pitch, but a solid thump as that wall of sound hits you.
If only because of commercial razor advertising, you’ve probably heard the word “mach” before. Mach number is a ratio between travelling speed and the speed of sound. Moving exactly at the speed of sound is called Mach 1. Traveling twice as fast is Mach 2, and so on. Military aircraft routinely travel at speeds greater than the speed of sound, though historically it was referred to as “breaking the sound barrier,” in reference to that pressure wave, which pilots report being very much like a physical wall, often producing very turbulent rides until it is passed. Chuck Yeager was the first person to fly faster than sound in 1947.
When the source passes beyond the speed of sound (supersonic), it begins to lead the wavefront. That means that you will see the plane pass overhead well before you hear the sounds it creates. In addition, as you can see from the animation below, the wavefronts being emitted are no longer just creating a single wall of pressure, but a cone of build-up that projects back along the path it has traveled. This feature is called a Mach Cone, and the angle that it comes off at depends on the mach number. It can be quite loud as it spreads out in all three dimensions.Equally as impressive as the supersonic boom produced is the sudden visible vapor cloud that forms around the aircraft as it makes the transition into supersonic speeds. Without delving too much into nonlinear dynamics and chaos, the basic explanation for this occurrence is that pressure perturbations in the air are suddenly amplified, which causes the rapid condensation of the water vapor.
Other things that use the Doppler Effect
It turns out that all waves can participate in the Doppler Effect, not just sound waves. When we talk about visible light, the color is determined by the frequency. The highest end of the frequency spectrum is blue light, the lowest is red light. So when a source of light – like a distant star – is traveling away from us here on earth, the Doppler Effect says we should see the light as red. Those moving towards us look blue. In fact, it is because distant stars show red shifts that we know the universe is expanding.
The echocardiogram – an imaging system of the heart and blood flow – also uses the Doppler shift. Doppler weather radars send out microwave radiation to measure light scattering off of small water droplets, measuring the shift in the waves that are bounced back at the detector. Police radars send out wavelengths of light that hit your car and bounce off. The reflected wave will have a Doppler shift based on your velocity, which the gun will convert to calculate your speed.
There’s also an instrumental rock group of the same name.
There isn’t much more to say
So that’s the Doppler effect. Talk about it with your friends if you want to show off your science-knowledge, it’s relatively easy to understand but there are enough details in it to sound flashy. Other than the occasional 19th century name-drop however, it’s a well-understood, well utilized, and well recognized feature of our day-to-day life. But beyond that, I don’t have anything more insightful to add.
If you’d like to improve the quality of my topic selection, feel free to email me at firstname.lastname@example.org. I won’t take offense...I promise.