A source moves at \(680\,\mathrm{m\,s^{-1}}\) where sound speed is \(340\,\mathrm{m\,s^{-1}}\). Find the Mach number.

A source has Mach number \(2.0\). Find the Mach cone half-angle.

A source moves at \(300\,\mathrm{m\,s^{-1}}\) where sound speed is \(340\,\mathrm{m\,s^{-1}}\). Is it subsonic or supersonic?

A source has Mach number \(1.25\). Is a shock cone expected in the simple model?

An aircraft flies at Mach \(3.0\). Find the Mach cone half-angle.

A shock cone half-angle is \(30^\circ\). Find the Mach number.

A source moves at Mach \(1.5\) in air where \(v=340\,\mathrm{m\,s^{-1}}\). Find its speed.

An aircraft flies at \(510\,\mathrm{m\,s^{-1}}\) where \(v=340\,\mathrm{m\,s^{-1}}\). Find its Mach number and Mach angle.

At a high altitude the sound speed is \(295\,\mathrm{m\,s^{-1}}\). What speed corresponds to Mach \(2.2\)?

A source has \(M=1.01\). Is the Mach angle close to \(90^\circ\) or close to \(0^\circ\)? Explain.

A supersonic aircraft at altitude \(2400\,\mathrm{m}\) has Mach angle \(45^\circ\). Estimate the horizontal distance behind the aircraft where the shock reaches the ground, assuming straight-line geometry.

A source moving at \(600\,\mathrm{m\,s^{-1}}\) creates a Mach angle of \(34^\circ\). Estimate the sound speed in the medium.

Compare the Mach angles for \(M=1.5\) and \(M=3.0\). Which shock cone is narrower?

A source travels \(900\,\mathrm{m}\) in the same time that one emitted wavefront expands \(300\,\mathrm{m}\). Find the Mach number and cone angle.

Derive \(\sin\theta=1/M\) using the distance traveled by the source and the radius of a sound wavefront after time \(t\).

Show that no real Mach cone angle exists for \(M<1\), and interpret the mathematical result physically.

An aircraft moves horizontally at speed \(v_S\) at altitude \(h\), with Mach angle \(\theta\). Derive the time delay between the aircraft passing overhead and the sonic boom reaching an observer directly below the path.

A source accelerates from subsonic to supersonic motion. Explain what changes in the wavefront geometry as it crosses \(M=1\), and why the ordinary subsonic Doppler formula stops describing sound ahead of the source.

A Mach cone half-angle is measured as \(\theta\) with uncertainty small enough to ignore. Derive \(v_S\) in terms of \(\theta\) and sound speed \(v\), then determine how \(v_S\) changes as \(\theta\) decreases.

Two supersonic sources move through the same medium with Mach numbers \(M_1\) and \(M_2=2M_1\). Derive the relationship between their Mach angles and explain why the angle does not halve in general.