• 6-8: 4F/M4. Vibrations in materials set up wavelike disturbances that spread away from the source. Sound and earthquake waves are examples. These and other waves move at different speeds in different materials.
• 9-12: 4F/H6ab. Waves can superpose on one another, bend around corners, reflect off surfaces, be absorbed by materials they enter, and change direction when entering a new material. All these effects vary with wavelength.
• 9-12: 4F/H6c. The energy of waves (like any form of energy) can be changed into other forms of energy.

• 9-12: 9C/H3c. A graph represents all the values that satisfy an equation, and if two equations have to be satisfied at the same time, the values that satisfy them both will be found where the graphs intersect.

• 6-8: 11B/M4. Simulations are often useful in modeling events and processes.
• 6-8: 11B/M5. The usefulness of a model depends on how closely its behavior matches key aspects of what is being modeled. The only way to determine the usefulness of a model is to compare its behavior to the behavior of the real-world object, event, or process being modeled.

• A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
• A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

• F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.^?
• F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

• F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

• F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^?

This model simulates the wave-generating machine created by John Shive at Bell Laboratories and made famous by the Similarities in Wave Behavior film. The simulation produces sinusoidal waves by twisting the first rod with a given frequency and amplitude.  Students can set the end to be free or fixed, select a function for twisting the first rod, add a damping force, or change the length of the bars to visualize how a wave propagates in a non-uniform medium. Graphs are displayed that plot mechanical energy as a function of frequency. Advanced students can compute the speed of the traveling waves at various frequencies and plot dispersion curves.

• Algebra-Based Physics Standing Waves and Resonance Unit
• AP/Calculus-Based Physics Standing Waves and Resonance Unit