Physci text 10 Waves

This material is often introduced with a demonstration of a system that produces a wave and the terminology used with respect to wave features.

Equipment often used in wave concept demonstrations

•     Super ball or other high bounce ball such a space ball.

•     Super ball or other high bounce ball such a space ball.

•     Stopwatch

•     Very light chain

•     Meter stick

•     Calculator

An introduction to the concepts of oscillations

A ball being bounced is a good example of a repeating system. In physics, repeating systems are described by special terms. When the ball repeatedly returns to a previous position and velocity, each repeat is called a cycle. A system that repeats is also known as an oscillating system.

The duration in time for one cycle is called a period τ. Period is often calculated by timing many cycles. Divide the time by the number of cycles to get the period. The units for period are seconds per cycle.

The mathematical reciprocal, dividing the number of cycles by the time, is called the frequency f. The units for frequency are cycles per second. Cycles per second has a special name: Hertz. One Hertz is one cycle per second.

If the distance the ball is dribbled is small (short), the then frequency f is high.

If the distance the ball is dribbled is large (long), the then frequency f is low.

Instructional note: Demonstrate using a stop watch, timing ten to twenty cycles (dribbles). Calculate both the period and frequency.

Waves on a rope

The distance from one yellow circle to the next circle on the diagram above is one wavelength λ. In the above diagram one wavelength is 48 centimeters. The symbol λ is the lower case Greek letter "lambda." Wavelength is a space measurement.

The distance from the middle of the wave to the crest, triangle to triangle, is the amplitude a. In the above diagram the amplitude a is 20 centimeters. Amplitude is a space measurement.

In the diagram above:

•     Wave A has an amplitude a of 5 cm

•     Wave B has an amplitude a of 10 centimeters

•     The wavelength λ of wave A is 20 cm.

•     The wavelength λ of wave B is 8 cm.

Amplitude and wavelength are measures of space. Waves have a time component as well. The time component of a wave is the frequency.

    The number of wavelengths (think: crests) that pass a point per unit time is the frequency f. The frequency is also the rate in cycles per second for the wave - the same concept as further above. Frequency is a time measurement and is not shown in the graphs above.

    The generic equation of a basic wave is y = sin(x) where sin is the trigonometric function sine. The actual function for the equation in the first graph in this section is:

y=sin(2πx÷48)

In 1864 James Clerk Maxwell wrote, "For the sake of persons of different types of mind, scientific truth should be presented in different forms and should be regarded as equally scientific whether it appears in the robust form and vivid colouring of a physical illustration or in the tenuity and paleness of a symbolic expression." The Scientists, John Gribbin, page 429. Amazon.

Physics first studied the physical world that we can see, hear, and touch. As physics studied both larger systems - such as the universe, and smaller systems, such as atoms, the physical models imagined became more of a mental picture our mind can comprehend than the actual reality. A theory called string theory imagines the universe consisting of what are mathematically not unlike vibrating strings, yet the reality - if string theory is true - is simply beyond our imagination. The only "reality" - if any - are the mathematical equations that describe the system. Nature is mathematical.

Wave velocity

When time and space combine linearly, the result may be a velocity. Waves have a velocity. The velocity v for a wave is calculated by multiplying the wavelength λ by the frequency f.

v = λ ƒ

Velocity of a water wave in shallow water where g is the acceleration of gravity and d is the depth of the water.

v=√gd

Instructional notes: Using a meter stick and stopwatch demonstrate the relationships between frequency f and wavelength λ. Diagram the wavelength and amplitude on the board. Calculate the velocity v of a wave on a chain. Cover the connections to sound waves, water waves. 

Laboratory 10: Measuring the speed of sound

Questions

•     What is the speed of sound?

•     Is the relationship a linear relationship between the time and distance for sound? That is, does sound slow down (or speed up) with increasing distance?

Equipment

•     Wood clappers such as two by fours

•     Stopwatches

•     Distance measuring equipment

•     gloves for the clapper

Instructional note: This lab may take quite a bit of time to gather data in between rain showers. Rain makes hearing the sound difficult. The experiment can usually only proceed in between rain showers.

Laboratory objective: To measure the speed of sound (also known as Mach one).

Procedure

Direct measurement over distance

Where there is the linear space to utilize this procedure, this procedure produces the most accurate data. Speed of sound values generated are usually within 5% and can be within 1% of the actual speed of sound on that day. Note that the use of the median for 25 or more time measurements is part of the reason this procedure produces such accurate results. Hand or arm signals enhanced by a towel can be used to communicate. A spinning towel is often used to tell a clapper to "clap again." If available, the ability to use cell phones is a plus when more complex communications are necessary. To direct a unit to take cover due to weather, cell phones are often useful.

•     A clapper with easily visible boards stands at least 200 meters from a group of at least five listeners each with a stopwatch

•     The listeners time the delay between seeing and hearing the boards clap. All times are recorded.

•     The boards are clapped at least multiple times at each distance.

•     Determine and record the median time.

•     The listeners move further away by 50 meters and repeat the sets of measurements.

•     Continuing increasing the distance to the extent possible according to the geography. Boards may be visible up to 550 meters depending on conditions.

Data table consists of median times versus distance in meters.

Graph

For the direct measurement, time t₁ will be on the x-axis, and the distance d₁ will be on the y-axis.

Analysis

The data is likely to be linear. Use the relationship d₁~vt₁ in Desmos to obtain the speed of sound v.

The graph is a time (duration) versus space (distance) graph, the slope is the speed of sound.

The speed of sound varies primarily as a function of temperature. Use the temperature to determine the speed of sound in air at the time of the laboratory. To make this calculation look up the published speed of sound in air using a site such as HyperPhysics after the laboratory session is over. Cite your source using either APA or MLA.

Perform an error analysis on the experimental value for the speed of sound from your slope calculations versus the published speed of sound. Calculate a percentage error as well: (experimentally measured − published)/published. Do NOT forget the parentheses.

Discussion

Write up a discussion of the laboratory including a discussion of the results, the theoretic predictions, whether the experimental data agrees with the theory and how good that agreement might be, and potential sources of error. Anything less than 5% is excellent agreement, less than 10% is good agreement.

Resonance tube variation

On rainy days a resonance tube alternative is deployed using very large graduated cylinders and tuning forks. 

Tuning forks and resonance tubes

Tall graduated cylinder with a reasonably large opening.

Note that the fundamental tone resonance length is roughly one quarter of the wavelength.

Desmos is necessary to this laboratory as the relationship is a regression to an inverse. One could invert the wavelength and perform a linear regression, this would also necessitate inverting the resulting slope. Students would be unlikely to understand why this was being done. Desmos has no difficulty with the inverse equation regression.