There are two possible problems.
Figure 1. The PSU has high ripple.
- Ripple on the DC may make the circuit noisy. The datasheet doesn't say what the frequency of the ripple is. The PSU is switched mode as, judging by the shape, there isn't enough room for a regular transformer in it. (Almost all of these PSUs are SMPS these days.) Since these switch at high frequencies, > 20 kHz, the ripple may be inaudible.
simulate this circuit – Schematic created using CircuitLab
Figure 2. (a) Two 9 V SMPS supplies represented by 7809 9 V regulators. This should work. (b) Trying to generate a split rail supply using an 18 V supply and a 9 V regulator doesn't always work.
- Be aware that your power supplies can only source current from the positive terminal Your proposal looks OK but to show a potential problem that has caught some people out, have a look at 1b. In this case an 18 V supply has been used and a single 9 V regulator to generate the mid-supply "GND" rail. Now if we try to connect R1 as shown U4 can't sink the current and strange things will happen.
Your circuit should be OK. If ripple proves to be a problem a couple of hundred µF of capacitance across each rail should help.