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2.5: Fourier Analysis and Synthesis
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/2%3A_Combining_Waves/2.5%3A_Fourier_Analysis_and_Synthesis
You can also see the magnitude of the amplitude by holding the mouse the mouse button down and moving the mouse to the top of one of the peaks on the graph on the left. Here $t$ is time, $n$ is th...You can also see the magnitude of the amplitude by holding the mouse the mouse button down and moving the mouse to the top of one of the peaks on the graph on the left. Here $t$ is time, $n$ is the number of the harmonic or mode ( $n=1$ for the fundamental, $2$ for the second harmonic etc.), $A_{n}$ is the amplitude of harmonic or mode number $n$ and $f$ is the fundamental frequency ( $f=1/T$ ).
Waves: An Interactive Tutorial (Forinash and Christian)
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)
This is a set of interactive tutorials designed to teach the fundamentals of wave dynamics. It starts with very simple wave properties and ends with an examination of nonlinear wave behavior. The emph...This is a set of interactive tutorials designed to teach the fundamentals of wave dynamics. It starts with very simple wave properties and ends with an examination of nonlinear wave behavior. The emphasis here is on the properties of waves which are difficult to illustrate in a static textbook figure. The tutorial may be used in conjunction with a text or as a stand alone introduction to waves. Exposure to calculus and basic physics is assumed in the latter sections.
1.1: Sine Wave
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/1%3A_Basic_Properties/1.1%3A_Sine_Wave
Go back to the original wave by clicking the reload button, . Pause the wave and measure the wavelength, $\lambda$ , on the graph (find the $x$ location of two successive peaks or troughs using the...Go back to the original wave by clicking the reload button, . Pause the wave and measure the wavelength, $\lambda$ , on the graph (find the $x$ location of two successive peaks or troughs using the cursor; the wavelength is the $x$ distance between peaks or troughs).
2.2: Interference
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/2%3A_Combining_Waves/2.2%3A_Interference
For destructive interference $d\sin\theta_{\text{dark}}=(m+1/2)\lambda$ where $m=0,\:\pm 1,\:\pm 2,\:\pm 3\ldots$ In both cases $d$ is the distance between the center of the openings (the separa...For destructive interference $d\sin\theta_{\text{dark}}=(m+1/2)\lambda$ where $m=0,\:\pm 1,\:\pm 2,\:\pm 3\ldots$ In both cases $d$ is the distance between the center of the openings (the separation of the sources), the angle $\theta$ is the angle from the central maximum out to a minimum or maximum and $m$ numbers the maximums (or minimums) starting from the center ( $m=0$ ).
4.9: Solitons
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/4%3A_Applications/4.9%3A_Solitons
One physical system represented by the Sine Gordon equation is a stretched string which starts out on one end in one trough but at the location of the kink goes over the hump to lie in the neighboring...One physical system represented by the Sine Gordon equation is a stretched string which starts out on one end in one trough but at the location of the kink goes over the hump to lie in the neighboring trough. Using the physical analogy of the elastic string in a series of troughs, what is the difference between the solution with the plus sign (the kink) and the solution with the minus sign (called an anti-kink)?
3.2: Collisions with Boundaries
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/3%3A_External_Interactions/3.2%3A_Collisions_with_Boundaries
Waves reflect from a boundary in two basic ways depending on whether the boundary is "hard" or "soft". In the case of waves on a string a "hard" boundary is where the string is firmly attached and a "...Waves reflect from a boundary in two basic ways depending on whether the boundary is "hard" or "soft". In the case of waves on a string a "hard" boundary is where the string is firmly attached and a "soft" boundary is when the end of the string can slide up and down.
4.3: Antenna
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/4%3A_Applications/4.3%3A_Antenna
For a receiving antenna oriented in the $y$ -direction, an oscillating field traveling in the $x$ -direction will cause the charges in the receiver to oscillate in the $y$ -direction with the same ...For a receiving antenna oriented in the $y$ -direction, an oscillating field traveling in the $x$ -direction will cause the charges in the receiver to oscillate in the $y$ -direction with the same frequency as the wave (the charges cannot move in the $x$ -direction because they are confined to the wire).
4.8: Non-Linear Waves
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/4%3A_Applications/4.8%3A_Non-Linear_Waves
The right side of the equation is basically the vertical acceleration of a piece of the string and the left side is the force. If the energy starts to drift significantly this is a sign that the numer...The right side of the equation is basically the vertical acceleration of a piece of the string and the left side is the force. If the energy starts to drift significantly this is a sign that the numerical calculations of the simulation are failing and the simulation no longer represents the elastic string (or any real system).
3.7: Impedance
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/3%3A_External_Interactions/3.7%3A_Impedance
The kind of wave reflected and the amount of energy transmitted depend on the properties of the material on either side of the boundary. In this simulation the change in mass of the string affects the...The kind of wave reflected and the amount of energy transmitted depend on the properties of the material on either side of the boundary. In this simulation the change in mass of the string affects the amount of reflected and transmitted energy and the speeds of the waves. This is because there is an impedance missmatch for waves inside the instrument (where the pressure is constrained by the sides of the instrument) and the pressure outside.
2.3: Group Velocity
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/2%3A_Combining_Waves/2.3%3A_Group_Velocity
In cases where several waves add together to form a single wave shape (called the envelope) we can quantify the speed with two numbers, the group velocity of the combined wave and the phase velocity. ...In cases where several waves add together to form a single wave shape (called the envelope) we can quantify the speed with two numbers, the group velocity of the combined wave and the phase velocity. For $k_{2}=8.4\text{ rad/m}$ , $\omega_{2}=8.4\text{ rad/s}$ try several values of $\omega_{1}$ and $k_{1}$ such that the ratio $\omega_{1}/k_{1}$ is always equal to one (the same as $\omega_{2}/k_{2}$ ).
1.2: Speed of a Wave
https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_(Forinash_and_Christian)/1%3A_Basic_Properties/1.2%3A_Speed_of_a_Wave
Determine the speed of the wave in the simulation using $v=\lambda /T$ where wavelength and period are determined from the simulation as you did in the previous exercise using the mouse to find the ...Determine the speed of the wave in the simulation using $v=\lambda /T$ where wavelength and period are determined from the simulation as you did in the previous exercise using the mouse to find the wavelength and the time to find the period.

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