(a) Suppose a sound wave is emitted from the origin at time...

(a) Suppose a sound wave is emitted from the origin at time 0 . After $t$ seconds ( $t>0$ ), explain why the position in units of the sound wave is modeled by $x=t \cos \theta$ and $y=t \sin \theta$, where the dummy parameter $\theta$ has range $0 \leq \theta \leq 2 \pi$. (b) Find parametric equations for the position at time $t$ seconds $(t>0)$ of a sound wave emitted at time $c$ seconds from the point $(a, b)$ (c) Suppose that a jet has speed $0.8$ unit per second (i.e., Mach $0.8$ ) with position function $x(t)=0.8 t$ and $y(t)=0$. To model the position at time $t=5$ seconds of various sound waves emitted by the jet, do the following on one set of axes. (1) Graph the position after 5 seconds of the sound wave emitted from $(0,0)$; (2) graph the position after 4 seconds of the sound wave emitted from $(0.8,0)$; (3) graph the position after 3 seconds of the sound wave emitted from $(1.6,0)$; (4) graph the position after 2 seconds of the sound wave emitted from $(2.4,0)$; (5) graph the position after 1 second of the sound wave emitted from $(3.2,0)$; (6) mark the position of the jet at time $t=5$. (d) Repeat part (c) for a jet with speed $1.0$ unit per second (Mach 1). The sound waves that intersect at the jet's location are the "sound barrier" that must be broken. (e) Repeat part (c) for a jet with speed 1.4 units per second (Mach 1.4). (f) In part (e), the sound waves intersect each other. The intersections form the "shock wave" that we hear as a sonic boom. Theoretically, the angle $\theta$ between the shock wave and the $x$-axis satisfies the equation $\sin \theta=\frac{1}{m}$, where $m$ is the Mach speed of the jet. Show that for $m=1.4$, the theoretical shock wave is formed by the lines $x(t)=$ $7-\sqrt{0.96} t, y(t)=t$ and $x(t)=7-\sqrt{0.96} t, y(t)=-t$. Superimpose these lines onto the graph of part (e). (g) In part (f), the shock wave of a jet at Mach $1.4$ is modeled by two lines. Argue that in three dimensions, the shock wave has circular cross sections. Describe the three-dimensional figure formed by revolving the lines in part (f) about the $x$-axis.

(a) Suppose a sound wave is emitted from the origin at time 0 . After $t$ seconds ( $t>0$ ), explain why the position in units of the sound wave is modeled by $x=t \cos \theta$ and $y=t \sin \theta$, where the dummy parameter $\theta$ has range $0 \leq \theta \leq 2 \pi$.
(b) Find parametric equations for the position at time $t$ seconds $(t>0)$ of a sound wave emitted at time $c$ seconds from the point $(a, b)$
(c) Suppose that a jet has speed $0.8$ unit per second (i.e., Mach $0.8$ ) with position function $x(t)=0.8 t$ and $y(t)=0$. To model the position at time $t=5$ seconds of various sound waves emitted by the jet, do the following on one set of axes. (1) Graph the position after 5 seconds of the sound wave emitted from $(0,0)$; (2) graph the position after 4 seconds of the sound wave emitted from $(0.8,0)$; (3) graph the position after 3 seconds of the sound wave emitted from $(1.6,0)$; (4) graph the position after 2 seconds of the sound wave emitted from $(2.4,0)$; (5) graph the position after 1 second of the sound wave emitted from $(3.2,0)$; (6) mark the position of the jet at time $t=5$.
(d) Repeat part (c) for a jet with speed $1.0$ unit per second (Mach 1). The sound waves that intersect at the jet's location are the "sound barrier" that must be broken.
(e) Repeat part (c) for a jet with speed 1.4 units per second (Mach 1.4).
(f) In part (e), the sound waves intersect each other. The intersections form the "shock wave" that we hear as a sonic boom. Theoretically, the angle $\theta$ between the shock wave and the $x$-axis satisfies the equation $\sin \theta=\frac{1}{m}$, where $m$ is the Mach speed of the jet. Show that for $m=1.4$, the theoretical shock wave is formed by the lines $x(t)=$ $7-\sqrt{0.96} t, y(t)=t$ and $x(t)=7-\sqrt{0.96} t, y(t)=-t$. Superimpose these lines onto the graph of part (e).
(g) In part (f), the shock wave of a jet at Mach $1.4$ is modeled by two lines. Argue that in three dimensions, the shock wave has circular cross sections. Describe the three-dimensional figure formed by revolving the lines in part (f) about the $x$-axis.

Key Concepts

Three-Dimensional Shock Geometry

In three dimensions, the shock wave generated by a supersonic object forms a cone with circular cross sections. This is achieved by revolving the lines or curves that describe the two-dimensional shock boundaries around the object's line of travel, typically resulting in a conical structure that encapsulates the region affected by the shock wave.

Shock Wave Formation and Sonic Boom

When an object travels faster than the speed of sound, its emitted wavefronts accumulate and intersect, creating a shock wave. This result is observed as a sonic boom. The shock wave is often characterized by a specific Mach angle, determined by the geometry of the intersecting circles or spheres, and provides insight into the boundaries of the shock front.

Mach Number and Relative Speed

The Mach number is the ratio of an object's speed to the speed of sound in the medium. It plays a crucial role in understanding how sound waves interact with moving objects. For instance, when an object moves at subsonic speeds (Mach less than one), its emitted sound waves always remain ahead of it, but at supersonic speeds (Mach greater than one), the waves can overlap to form distinct phenomena like shock waves.

Parametric Equations

Parametric equations are a way to represent curves by expressing the coordinates as functions of a parameter. In the context of sound waves, using equations like x = t cos? and y = t sin? describes a circle with radius equal to t, capturing the idea that sound propagates equally in all directions from the source over time.

Circular (or Spherical) Wave Propagation

Sound waves emitted from a point source spread out uniformly in all directions. In two dimensions, these wavefronts form circles whose radii increase linearly with time, while in three dimensions they form spheres. This concept underlies the use of polar coordinates to represent the position of a sound wave after a given time.

Translation of Wave Source and Time Shift

When a sound wave is emitted from a source that is not located at the origin or is emitted at a time other than zero, the wave's position must be adjusted accordingly. This involves translating the parametric equations by the source’s coordinates and incorporating the time delay, so that the wavefront is correctly positioned relative to the moving source.

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