x shock ¼
2c
2
0
γ þ 1
ð
Þa
:
ð2:27Þ
Returning to Eq. (2.25) and writing it in the form,
2ax ¼ À2c 0 u À γu
2
þ 2at c 0 þ
γ þ 1
2
u
h
i
,
which can be regarded as a quadratic equation in u with t as a parameter, so that,
γu
2
þ 2 c 0 À
γ þ 1
2
at
h
i
u À 2a c 0 t À x
ð
Þ¼0:
Hence, the following expression for u is obtained,
u ¼
À c 0 À
γþ1
2
À Á
at
Â
à þ c 0 À
γþ1
2
À Á
at
È
É 2 þ 2γa c 0 t À x
ð
Þ
h
i 1=2
γ
,
ð2:28Þ
which is only a valid solution up to the time that the shock wave is formed. We can
see from the latter equation that u ¼ 0 when x ¼ c 0 t. In addition,
u ¼
2γa c 0
2c 0
γþ1
ð
Þa À x
h
i 1=2
γ
when t ¼
2c 0
γþ1
ð
Þa , which is the time that the shock wave forms. The position of the
shock front is
x shock ¼
2c
2
0
γ þ 1
ð
Þa
,
which is at the forward front of the wave, that is, at x ¼ c 0 t, where t ¼ 2c 0 /(γ + 1)a
according to Eq. (2.26). In Fig. 2.8 we show plots of the particle velocity u as a
function of position x according to Eq. (2.28) for three different times t. For the plots
the following parameters were assumed; γ ¼ 1.4, c 0 ¼
ffiffiffiffiffiffi ffi
1:4
p
and a ¼ 0.01. Using
these parameters in Eqs. (2.26) and (2.27), one can see that the shock wave will form
at t ¼ 98.6 (arb. units), and the place of formation is at x ¼ 116.7(arb. units) which
corresponds to the forward front of the wave. It can be observed that the gradient
(∂u/∂x) becomes infinite at x ¼ 116.7. Figure 2.8 can be compared with similarly
generated plots obtained later on in Chap. 4 where the equations are numerically
integrated with artificial viscosity included.
56
2 Waves of Finite Amplitude
2c
2
0
γ þ 1
ð
Þa
:
ð2:27Þ
Returning to Eq. (2.25) and writing it in the form,
2ax ¼ À2c 0 u À γu
2
þ 2at c 0 þ
γ þ 1
2
u
h
i
,
which can be regarded as a quadratic equation in u with t as a parameter, so that,
γu
2
þ 2 c 0 À
γ þ 1
2
at
h
i
u À 2a c 0 t À x
ð
Þ¼0:
Hence, the following expression for u is obtained,
u ¼
À c 0 À
γþ1
2
À Á
at
Â
à þ c 0 À
γþ1
2
À Á
at
È
É 2 þ 2γa c 0 t À x
ð
Þ
h
i 1=2
γ
,
ð2:28Þ
which is only a valid solution up to the time that the shock wave is formed. We can
see from the latter equation that u ¼ 0 when x ¼ c 0 t. In addition,
u ¼
2γa c 0
2c 0
γþ1
ð
Þa À x
h
i 1=2
γ
when t ¼
2c 0
γþ1
ð
Þa , which is the time that the shock wave forms. The position of the
shock front is
x shock ¼
2c
2
0
γ þ 1
ð
Þa
,
which is at the forward front of the wave, that is, at x ¼ c 0 t, where t ¼ 2c 0 /(γ + 1)a
according to Eq. (2.26). In Fig. 2.8 we show plots of the particle velocity u as a
function of position x according to Eq. (2.28) for three different times t. For the plots
the following parameters were assumed; γ ¼ 1.4, c 0 ¼
ffiffiffiffiffiffi ffi
1:4
p
and a ¼ 0.01. Using
these parameters in Eqs. (2.26) and (2.27), one can see that the shock wave will form
at t ¼ 98.6 (arb. units), and the place of formation is at x ¼ 116.7(arb. units) which
corresponds to the forward front of the wave. It can be observed that the gradient
(∂u/∂x) becomes infinite at x ¼ 116.7. Figure 2.8 can be compared with similarly
generated plots obtained later on in Chap. 4 where the equations are numerically
integrated with artificial viscosity included.
56
2 Waves of Finite Amplitude